•%*
^.
^%:
'>^
^ «> ^' nO.
i?
/i
<«
/
°\ '\'
^>%
!>'
IMAGE EVALUATION
TEST TARGET (MT3)
1.0
IM 1125
I.I
Hi
2.0
1.25
U IIIIII.6
PhntncrmnViir
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14530
(716) 87^4503
1
^
^/^
f/
«
,,G^'
fV
^v
:\
\
^9)
V
<>
<«>.
6^
^v
»
IS 4b
CIHM/ICMH
Microfiche
Series.
CIHM/ICMH
Collection de
microfiches.
Canadian institute for Historical Microreproductions / Institut Canadian de microreproductions historiques
Technical and Bibliographic Notes/Notes techniques et bibliographiques
The Institute has attempted to obtain the best
original copy available for filming. Features of this
copy which may be bibliographically unique,
which may alter any of the images in the
reproduction, or which may significantly change
the usual method of filming, aw checked below.
D
Coloured covers/
Couverture de couieur
I I Covers damaged/
Couverture endommagi^e
Covers restored and/or laminated/
Couverture restaurde et/ou pelliculde
Cover title missing/
Le titre de couverture manque
Coloured maps/
Cartes gdographiques en couieur
Coloured ink (i.e. other than blue or black)/
Encre da couieur (i.e. autre que blaue ou noire)
Coloured plates and/or illustrations/
Planches et/ou illustrations en couieur
Bound with other material/
Relid avec d'autres documents
D
n
n
Tight binding may cause shadows or distortion
along interior margin/
Lareliure serree peut causer de I'ombre ou de la
distorsion le long de la marge intdrieure
Piiank leaves added during restoration may
appear within the text. Whenever possible, these
have been omitted from filming/
II se peut que certaines pages blanches ajout^es
lors dune restauration apparaissent dans le texte,
mais, lorsque cela 6tait possible, ces pages n'ont
pas 6ti filmdes.
Additional comments:/
Commentaires suppldmentaires;
L'Institut a microfilm^ le meilleur exemplaire
qu'il lui a 6t6 possible de se procurer. Les details
de cet exemplaire qui sont peut*tre uniques du
point de vue bibliographique, qui peuvent modifier
une image reproduite, ou qui peuvent exiger une
modification dans la m^thode normale de filmage
sont indiquds cidessous.
r~n Coloured pages/
Pages de couieur
Pages damaged/
Pages endommag^es
Pages restored and/or laminated/
Pages restaurdes et/ou pelliculdes
Pages discoloured, stained or foxed/
Pages ddcolor^es, tachetees ou piquees
Pages detached/
Pages d^tachees
v/
Showthrough/
Transparence
Quality of print varies/
Quality indgale de {'impression
Includes supplementary material/
Comprend du materiel supplementaire
Only edition available/
Seule Edition disponible
D
Pages wholly or partially obscured by errata
slips, tissues, etc., have been refilmed to
ensure the best possible image/
Les pages totalement ou partieilement
obscurcies par un feuillet d'errata, une pelure,
etc., ont ^xi film^es i nouveau de facon i
obtenir la meilleure image possible.
1
a
1
V
IN
d
e
b
ri
n
n
This item is filmed at the reduction ratio checked below/
Ce document est film* au taux de reduction indiqud cidassous.
10X 14X 18X 22X
26X
30X
12X
16X
20X
24X
28X
32X
s
itails
s du
lodjfier
r une
Image
rrata
o
)elure,
3
32X
The copy filmed here hes b«3en reproduced thanks
to the generosity of:
D. B. Weldon Library
University of Western Ontario
(Regional History Room)
The images appearing here are the best quality
possible considering the condition and legibility
of the original copy and in keeping with the
filming contract specifications.
Original copies in printed paper covers are filmed
beginning with the front cover and ending on
the last page with a printed or illustrated impres
sion, or the back cover when appropriate. All
other original copies are filmed beginning on the
first page with a printed or illustrated impres
sion, and ending on the last page with a printed
or illustrated impression.
The last recorded frame on each microfiche
shall contain the symbol — ^^ (meaninot "CON
TINUED"), or the symbol V (meaning "END"),
whichever applies.
Maps, plates, chaits. etc., miy be filmed at
different reduction ratios. Those too large to be
entirely included in one exposure sire filmed
beginning in the upper left hand earner, left to
right and top to Jottom. as many frames as
required. The following diagrams illustrate the
method:
L'axemplaire film4 fut reproduit grdce ^ la
gAn^rositi de:
D. B. Weldon Library
University of Western Ontario
(Regional Histor/ Room)
Lee images suivantes ont 4t4 reproduites avec le
plus grand soin. compte tenu de la condition at
de la nettetA de I'exemplaire film^, at en
conformity avec las conditions du contrat de
filmage.
Las exemplaires originaux dont la couverture an
papier est imprim^ sont filmto en commen^ant
par le premier plat at en terminant soit par la
darni^re page qui comporte une empreinte
d'impression ou d'illustration, soit par le second
plat, selon le cas. Tous les autres exempiaires
originaux sont film^s en commenpant par la
premiere page qui comporte une empreinte
d'impression ou d'illustration et en terminant par
la derniire page qui comporte une telle
empreinte.
Un dtis symboles suivants apparaitra sur la
demidre image de cheque microfiche, selon le
caa: la symbole — ^ signifie "A SUIVRE", la
symbols V signifie "FIN".
Les cartes, planches, tableaux, etc., peuvent etre
filmte dt dee taux de rMuction diffdrents.
Lorsqua le document est trop grand pour etre
reproduit en un seul cliche, il est filmd d partir
da Tangle sup^rieur gauche, de gauche d droite,
et de haut en bas, en prenant le nombre
d"images n^cdssaire. Les diagrammes suivants
iilustrent la mdthode.
1
2
3
4
5
6
TJ
Aa
PI
Serica of National 0cl)ool Book
TREATISE ON ARITHMETIC,
m
THEORY AND PEAC'JICE.
^ I
FOR
^%^ UlSit (9(5 ^I^DjlUiVa^.
Aathorhd hy the Council of Public Instruction,
for Upper Canada.
TORONTO:
PUnLISIlI^^D BY ROBERT MclMIAIL,
65, KTiNG Strket East. •
1860. *
\
1 V
\
\
\n
ti
t€
\
''Vv^^' ij^^ //
/
I' K E F A U JK
>
In Iho prosont edition a vast number of eicrcisM I,.,.
be,n «ldod, that no ,„,e, however trifling nlTL ,oft
w.thout ,0 many ill„,t™tiona as should f rvet ml u
uftc,ently fi.^.har t. the pupi,. And when it was feared
mglit not at oneo suggest itself; some question calculated
ocrs and " the pnnoiples of notation and numeration 
for the teacher may re.t assured, that the facility, I^d e^^
the success, wUh which subsequent pa>ts of his tostrucZ
wirbe .onveycd to the mind of the learner, dependsinl'
great degree, upon an adequate acquaintance wHh ti
Hence, to proceed without scouring a perfect and pra^S
knowledge of this part of the suyect, is to retard'TaS
than to accelerate improvement.
. ?» '■"P''.*'™"' "« ^y oommencemcnt, must be m.>da
due d. Of the great utility of teehnica'. iangua..e faecut
tately „ndersto«l) it is almost superfluous to say anvZv
hero : we cannot, however fni.l,o..,. ^i ■ ' ""y'hmg
cillin,, t,,.. • ""we\oi, iMbear, upon this oocas on, re
call ng to ren,cmbra«ee wh.at is so admirably and so effeo.
J.ven n the common mchanical arts, something of a
t.chn,cal language is 6,md needful f„r th<L. who .arMcaru?
'* PREFACE ,
ing or exercising them. It would be a very groat la
convenicnpo, even to a common carpenter, not to have a
precise, well understood name for each of the several opera
tiona he performs, such as chiselling, sawing, planing, ^^c,
md for the several tools [or instruments] he works with.
And If we had not such words as addition, subtraction,
multiplication, division, &c., employed in an exactly defined
sense, and also fixed rules for conducting tho.e and other
arithmetical processes, it would be a tedious and uncertain
work to go through even such simple calculations as a child
very soon learns to perform with perfect ease. And after
all there would be a fresh difficulty in making other per
sons understand clearly the correctness of the calculations
made.
"You are to observe, however, that technical Ian.rua.^o
and rules, if you would make them really useful, must be not
only (bshndly urulerstood, but also learned and yemembered as
famiharly as the alphabet, and employed cmshmtlu, and
With Born^uhns eimtnm; otherwise, technical languae will
p.ove an encumbrance instead of an advantage, just as a
suit of clothes would b. if, instead of putting thorn on and
wearing them, you.were to carry thorn about in your hand "
Page 11.
What is said of technical lan^riMge is, at least, equally true
0. the signs and characters by which we still further tae'ilitate
the conveyance of our ideas on such matters as form the
subject of the present work. It is mucli more simple to put
down a character whicu expresses a process, than to write
the name, or description of the latter, in fuJl. Besides, in
glancing over a mathematical investigation, the mind is
able, with greater ease, to connect, and understand its dif
ferent portions when they are briefly expressed by familiar
Bigns, than when they are indicated by words vdnch havo
nothing particularly calculated to catch the eije, and which
cannot even be clearly undcrstocKi without considerable
attention. But it mu«c be bortie in mind, that, while such
«■ trsatise as the present, will seem easy and intelligible
▼a
PREFACE
enough if M,e „ign,, ^Uch it contains in almost evorr na™
mo or loss obscure to those who have not been habitfS
t he „so them. They are, however, so few and so L^H
that there is no eicnse for their not hoin^ perfcctlv onir
stoodpartienlarly by the teacher of arithme:;o ' '''■
hhould peculiar e.rcumstanccs render a different arrange
■ IT ' u ° ■"■''"'"' "' ■«"^'' ""i^^aWe, the j„dicio,«
master will never be at lo8« how to act^there «« Z
ntelhgcnco of tho pupil, will render it necessary to conflno
u .nstrucfon to the more important branches. 4, i^^r
Bhould, If possible, mako it an inviolable rule to receive
rerorVetr """"'"ri' "^ "^ e^pWion r
reason. The references which have been subjoined to tha
different questions and which indicate the paJgiphs wh re
neswrh "'' ''"^*'" be obtained, and alsfthL^r
enees which are scattered throagh the work, will, be fould
of considerable assistance ; for, as the most i^teUint pnTu
wi 1 oecasionally forget something he has learned he^a^
e.ple" ■ ""' '™ '»™"*^' *'"'™ '"e has seen it
I)«™<,& have been treated of at the same time as integers
be^^ause since both of them follow ,re,set, the same ifw"'
when the rales relating to integers are fullv understood
here IS „o&V,. „» to be learn:! on the sub/eelrpt^ ct
larly If what has been said with reference to numeratirand
notation is carefully borne in mind. Should it, however "n
unti the learner shall have made some further advance.
_ ilie most useful portions of menlai arithmetic have been
ntrodueed into "Practice and the other rules with whch
they seemed more immediately connected.
the mind of the toner and when he is found to have been
^" PRKFACE.
guilty of any inaccuracy, ^o Bhould be made to correct him
»elf by repeating each part of the appropriate rule, and
oxeniphfyin it, until ho perceives hia error. It should be
continually kept in view that, in a work on such a subject
as arithmetic, any portion must seem difficult and obscura
without a knowledge of what precedes it.
The table of logarithms and article on the subject, also
the table of squares and cubes, squarr. roots and cube roota
of numbers, which have been introduced at the end of the
work, will, It IS expected, prove very acceptable to the more
•dvanced arithmetician.
n
CONTENTS
f
PART I.
Multiplication Table, . . . ,
Tables of money, weights, and mcasuree,
Definitions, • . . .
Section I.— Notation and Numeration,
Arabic system of numbers,
lionian notation, .
' • • •
Skction II. ^Simple Addition,
To prove Addition,
Addition of quantities containing decimals,
Simple Subtraxjtion,
To prove Subtraction,
iSubtraction of quantities containing decimals,
Simple Multiplication, .
To multiply when neitlier multiplicand nor mul
tiplier exceeds 12, .
When the multiplicand exceeds 12,
To prove Multiplication,
To multiply quantities, when there are cyphers
or decimals,
When both multiplicand and multiplier ex
eee«^. 12,
To prove Multiplication,
To prove MultipHcation, by casting out the nines,
To multiply, 60 as to have a certain number of
decimal places in the product,
To multiply by a composite number,
~ by a number not composite.
To multiply by a number consisting of nines.
Simple Division, . .
a 2
Par*
1
8
12
14
18
20
83
37
45
It)
41)
53
57
57
CO
Gl
6r>
G7
70
72
73
74
78
I'll
CONTENTS
when tho divisor does not exceed 12,
Mividend 12 times the divisor,
♦1, V .J ".'" *^'^'^°'" ^*^* ""* "oeed 12, bufc
4:ot:s:r"*^^^
To divide when the dividend, divisor, or both'
contain cyphers or decimals,
 When the divisor exceeds 12,
To prove Division, . * '
To divide by a composite number,
^^^^^ ^L^ "'""^'''^ ^" ""^« J^« *han one
expressed by unity and one or more cyphers,
lofind the greatest common measure of nimberi
10 tnd their least common multiple
SECTION III.Reduction Descending, ' '
Reduction Ascending, . . * * *
To prove Reduction, . ^,'
The Compound Rules, . .' * '
' Compound Addition, . ,' ' '
Compound Subtraction,
Compound Multiplication, when the multiplier
does not exceed 12, . .
• When the multiplier exceeds 12 and ig
composite, . . «"iu is
—  When the multiplier is the sum of compo^
Bite numbers, . . ^
— When the multiplier is not composite, .'
Compound Division, when the divisor la abstract,
and does not exceed 12, .
■ When the divisor exceeds 12 and' is com.
pgsite.
When the divisor exceeds 12 and is not com
posite,
— When the divisor and di vid'end are both apl
plicate, but not of the same denomination • or
more than ono 'lenomination is found in either
or both,
80
85
86
89
94
05
98
90
lOZ
104
lOV
109
110
113
114
123
126
128
123
130
132
•
134
134
139
^
•i
80
85
86
89
94
9S
96
96
lOZ
104
lOV
109
110
113
114
123
^
//
{
/.
COMTKNTi
/v*
/
%(:tion IV.— Vulgar Fractions,
\ Toreducoun improper fraction to a mixed n
To reduce un integer to a fraction,
To reduce fractions to lower terms,
To find the value of a fraction in terms of a lower
denoiaination, .
To express one quantity as the fraction of another,
To add fractions having a common denominator,
To add fractions when their denominators are
different and prime to each other.
To add fractions having different denominators,
not all prime to each other.
To reduce a mixed number to an improper frac
tion.
To add mixed numbers, .
' • •
To subtract fractions which have a commoh
denominator, .
■ • • •
To subtract fractions which have not a commoa
denominator, . . . . ,
To subtract mixed numbers, or fractions from
mixed numbers,
To multiply a fraction and whole number together
To multiply one fraction by another,
To multiply a fraction, or mixed number by
mixed number,
To divide a fraction by a whole number,
To divide a fraction by a fraction,
To divide a whole number by a fraction,
To divide a mixed number by a whole number or
fraction.
To divide an 'nteger by a mixed number,
To divide a fraction or mixed number by a mixed
number,
' • • •
►Vheu the divisor, dividend, or both, are com.
pound, or complex fractions, .
Oeoimal Fractions, ....
To reduce a vulgar fraction to a decimal or to a
decimal fraction.
To reduce a decimal to a lower denomination,
/140
^ 142
142
143
143
144
145
146
147
148
149
150
150
152
153
154
156
156
158
158
159
159
160
162
163
163
CONTENTS.
To find at <me the decimal equivaJent to any
number of shillings, pence, &o., ^
Circulating Decimals, . * * *
To change a circulating decimal into its equi*
vaJent vulgar fraction, ^
When a vulgar fraction will give a finite decimal,'
Thenumber of decimal places in a finite decimal
The number of digits iu the period of a circulate
— When a circulating decimal will contain a
hnite part, . .
Contractions in multiplication a«d division, de
rived from the properties of fractions,
Section^ \.— Proportion, .
Natiu^ of ratios, . , ''
Nature of Proportion,
To fil *^' ^jthmetical mean of* two quantities,'
To find a fourth proporti. al, when the first term
isumty, . . ^ "*
— When .ho second or third term is imity,
To find the geometrical mean of two quantities,
Fr operties of a geometrical proportion, .
Rule of Simple Proportion,
— When the first and second terms are not of
.he sa:ne or contain different denominations,
— When the tbud term contains more than one
denomination, ...
If fractions or mixed numbJrs are found in
any of the terms,
Piule of Compound Proportion,
To abbreviate the process, ' .
PART n.
&CTI0N VI.~Practice,
To find aliquot parts, . . * *
To 2nd the price of one denomination, ttiat of a
higher being given, ,
rtfft
164
im
169
167
170
170
171
172
173
176
17T
179
181
184
184
184
185
185
190
192
195
202
204
203
209
21]
1
J
raff*
iny
•
1Q4
mt
•
169
•
169
•
a
•
167
il,
170
il,
170
e,
171
a
•
172
9
•
173
•
176
•
17r
k
179
>,
181
1
•
184
•
184
184
1
185
185'
190
192
195
202
204
209
21]
CONTENTS.
To find the price of more than one lower denomi
nation, • . . ,
To find the price of one higher denomination, .'
To find the price of more than one higher deno
mination,
Given the price of one denomination, to find that
of any number of another, .
When the price of any denomination is the aliquot
part of a shilling, to find the price of any num
ber of that denomination,
When the price of any denomination is the aU
quot part of a pound, to find the price of any
number of that denomination, .
When the complement of the price, but not the
price itself, is the aliquot part or parts of a
pound or shilling.
When neither the price nor its complement is the
aliquot part or parts of a pound or shilling.
When the price of «ach article is an even number
of shiliin./A to find the price of a number of
ai'ticieK
* • • •
When the price is an odd numbor and loss
than 20, . .
To find the price of a quantity reproscited by a
mixed number,
Given the price per cwt., to find that of cwt.,
qrs., &c.,
Given the price per pound, to find that of cwt.,*
^qrs., &c.,
Given the price per pound, to find that of a ton.
Given the price per ounce, to find that of ounces,
pennyweights, &c., .
Given the price per yard, to find that of yards,'
qrs., &«.,
To find the price of acres, roods, &o., .
Given the price per quart, to find thet of a hoc*
tead. ".
Given the price per quart, to find that of a tun,
ziu
Page
212
213
213
214
215
215
216
217
218
219
220
221
222
223
223
225
226
226
227
xiy
CONTENTS
Given the pri^e of one article in pence, to find
that of any number, .
Given wages per day, to find their amount per
year, . . ^ ^ .
Bills of parcels, . . ; • .
Tare and Tret, . ' * *
• • • .
Section Vll.Simple Interest.
To find the simple interest on any sum, for a year,
— When the rate per cent, consists of more
than one denomination,
To find the interest on any sum for years*
For years, months, &o.,
To find the interest on any sum, for a^y time,"
at .5, b, &c., per cent., .
When the rate, or number of years, or both,'
are expressed by a mixed number.
To find the interest for days, at five per cent., .*
To find the interest for days, at any other ratq, .
10 find the interest for months, afr 6 per cent ,
To find the interest of money left after one or
more payments.
Given the amount, rate, and time~to find the
principal,
Given the time, rate, and principalto find the
amount,
Given the amount, principal, and rate— to find
the time.
Given the amount, principal, and time,' to find
the rate.
Compound Interestgiven the principd, rate,'
f '.d time— to find the amount and interest.
To find the present worth of any sum.
Given the principal, rate, and amoJnlrlto find
the time.
Discount,
• • •
To find discount, • , ,
To compute Commission. lnsiirnTift« 'p««i,^„«»
Pajf«
227
223
229
233
237
238
239
239
240
i
242
1
243
i
244
M
244
i
246
1
248
• i^?M
249
1
249
1
250
1
251
1
256
1
257
1
260
^
261
1
nan
li
Pagf«
227
223
229
233
237
238
239
239
240
242
243
244
244
246
248
249
249
250
251
256
257
260
261
nan
CCNTENT8.
To find what insurance must be paid that, if the
goods lire lost, both their value and the insur
ance paid may be recovered, ..
Purchase of Stock,
Equation of Payments, .
Sjcction VIII.— Exchange, .
Tables of foreign money.
To reduce bank to current money,
To reduce current to bank money.
To reduce foreign to British money,
To reduce British to foreign money,
To reduce florins, &c., to pounds, &c., Flemish,
To reduce pounds, &o., Flemish, to florins, &c.,
Simple Arbitration of Exchanges,
Compound Arbitration of Exchanges,
To estimate the gain or loss per cent..
Profit and Loss, .
• • •
To find the gain or loss per cent..
Given the cost price and gain— to find the selling
price, . . . . ^
Given the gain or loss per cent., and the selling
price*to find the cost price, .
Simple Fellowship,
Compound Fellowship, . . , '
Barter,
Alligation Medial,
Alligation Alternate,
When a given amount of the mixture is re
quire,d,
When the amount of one ingredient is given.
Sjcction IX.— Involution, .
To raise a number to any power.
To raise a fraction to any power.
To raise a mixed number to any power,
Evolution,
To find the square root, .
When the square contains decimals.,
To find the square root of a fraction,
XV
Page
263
265
266
268
269
272
273
274
277
281
282
283
285
286
287
288
289
290
291
293
296
298
299
302
304
306
307
307
308
308
308
'^U
M2
XVI
CONTENTS.
I
To and the square root of a miiod number,
10 IincI the cube root,
'T7 Y^®" *^^ ^^^^^ contains decimals, '
lo find the cube root of a fraction.
To find the cube root of a mixed number,*
1 extract any root whatever,
To find the squares and cubes, the squmo and
cube roots of numbers, by the table, .
Logarithms, .
\tle, ^^' ^'^^''^^'^ "^ ^ given numbed by th
To find the logarithm of a fraction.
To find the logarithm of a mixed numbed,
log^iim, "^"'^^ ^^^;'^^'^^"^^"« *« ^ Siven
— If the given logarithm is not in the' table, *
lo multiply numbers by means of their loga*
To divide numbers by melns of their logarithms*
l^^t^^^^^^^^^P^^^^
To evolve a quantity by means ofits logarithm. '
SECTION X.— On Progression,
To find the sum of a series of terms in arUhmeti'
cal progression, •
In an arithmetical series given the extremes and
number of termsto find the common diS
ence, , ^
\^.t r """"^'^ '^ arithmet'ical m;ans,be:
tween two given numbers,
^'s fries,'"^ ^''''^'"^'' '"'"^ '^ '^"y '^^^'itl^neticai
^"cZ&'7f'"^ ''"'^' *S™ '^' ^^*^«*"^ and
common differencetofindthenumberof terms,
~r. '\*^' '"™ "^ "^« ««^i««' "^e number of
Tn ni' ''"" «^tremeto find the other, .
To find the sum of a series of terms in geometri
oal progression, . b^ometri
rage
313
313
315
316
31G
' 317
318
319
321
323
323
324
324
325
326
327
327
329
329
330
ooi
332
332
333
CONTENTS.
Ih a geoulGtrical series, given the extremes and
number of term&— to find the common ratio,
To find any number of geometrical means between
two quantities,
To find any particular term of a geometrical series,
In * geometrical series, given the extremes and
common ratio— to find the number of terms, .
In a geometrical series, given the common ratio,
the number of terms, and one extreme— to find
the other • . . .
Annuities, . .
* ■ •
To find the amount of a certain number of pay
ments in arrears, and the interest due on them,
To find the present value of an annuity,
When it is in perpetuity, .
To find the value of an annuity in reversion,
Position, .
5"ingle Position, .
Double Position, .
Miscellaneous exercises,
Table of Logarithms,
Table of squares and cubes, and of square and
cube roots, «...
Table of the amounts of £1, at compound interest]
Table of the amounts of an annuity of £1,
Table of the present values of an annuity of ^1,
Irish converted into British acres,
' Value of foreign money in British, . [
Tago
335
336
336
337
338
340
340
342
343
344
345
346
347
355
361
377
385
385
386
386
386
T5/ EATLSi: ON ARITHMETIC:
IM
THEORY AND PRACTICE.
ARITHMETIC.
PART I.
TABLES.
MULTIPLICATION TABLE.
i
Twice
1 are 2
2—4
3 * 6
48
5 — 10
6 — 12
7  14
8 — 16
9 — 18
10 — 20
11 — 22
12 — 24
8 times
1 are 3
2—6
8—9
12
15
18
21
24
9 — 27
10 — 80
n — 83
12 — 36
4 times
1 are 4
2—8
3 r 12
4 — 16
5 — 20
6—24
7—28
8—32
9
10
11
12
86
40
44
48
5 times
1 are 5
2 — 10
3 — 15
4 — 20
6 — 25
6 — 30
7 — 35
8 — 40
9 — 45
10 — 60
11 — 55
12 — 60
6 times
1 are 6
2—12
3 — 18
4 — 24
6 — 30
6 — 36
7—42
8 — 48
9 — 54
10 — 60
11 — 66
12
7'^
7 times
1 are 7
2 — 14
8—21
4 — 28
5—35
6—42
7—49
8 — 66
9 — 63
10 — 70
11 — 77
12 — 84
8 times
1
are 8
2
— 16
3
— 24
4
— 32
5
— 40
6
— 48
7
— 56
8
— 64
9
— 72
10
— 80
11
— 88
12
— 9S
9 times
1 are 9
2—18
3—27
4—36
6—45
6—64
7 — 63
8—72
9—81
10 — 90
11 — 99
12 — 108
10 times
1 are 10
2—20
3—30
4—40
6—60
6 
7 
8 
9—90
10 — 100
11 — 110
12 — 120
60
70
80
11 times
1 are 11
2—22
3—33
4—44
5 — 55
6—66
7—77
8—88
9—99
10 — 110
11 — 121
12 — 132
12 times
1 are 12
2—24
3—36
4 — 48 I
6 — 60 I
6 — 72 I
7 — 84 ,
8 — 96 i
9 — 108 !
10 — 120 \
11 — 132
12 — li4
It appears from tliis table, that the multiplication of tho
same two uumbers, m whatever order taken, produces tlio
SIGNS USED IN THIS TREATISE.
'
7 times i
1
are 7
2
— 14
3
— 21
4
— 28
5
— 35
6
— 42
7
— 49
8
— 66
9
— 63
10
— 70
n
— 77
12
— 84
12 times
1
are 12
2
 24
3
 86
4
— 48
6
— 60
e
— 72 ,
7
 84 .
8
— 96 ;
9
~ 108 !
— 120 I
1
— 132
2
 Ui
ion of tho
iucca tlw
^^ + the sign of addition; as 5+7, or 5 to bo addc4
tra7ted''froT4"^ "*'''°"™ ' "^ ^^' "' ^ '» ^' »»''.
muHiptVbfg"' "'"J"''''"''" i »» 8X9, or 8 to be
^^•^ the sign of division ; as 18+6, or 18 to be divided
ti,„ ft? vinculum, which is used to show that all
the quanfafe s united by it are to be oonsiderld as but
rbe^^n^r^^^^I^lral^e^f^p^HeltJ
thlts'lTssfh™*i.""'" "■"* ' '^ S>«^'er than J, and
thJ rati^of frZ/'i" '" '■f"".™' *"' 5:6, means
me ratio ot j to 6, and is read 5 is to 6
: : indicates the eqiiality'of ratios ; thus, 5 : 6 • • 7 • 8
means that there is the slm. relation bet'ween 5 'and 6
as between 7 and 8 ; and is read 5 is to 6 as 7 is ?o 8
y the radical sign. By itself, it is the sign of the
2:\t rootff 4', or 4^ it "^^ ^' '' '' '' ^^' '^
64131, &c. may be read thus : taE 3 f?om 8, add 7 to thJ
difference, multiply the sum by 4, divide the nroduct hv fi
take the square root of the quotient Td to °t ^dd 31 tLn
multiply the sum by the cube root of 9, divide tho product
^ — r' — '•' ••' product will bo equal to 04131. &c
1 hesc «sns arc /tUly cq^laincd in tlair proper places.
I
MULTIl'LICATIOiN TAIJI.E.
eaino result; tlius I times 0, and G times 5 jiro 30:— tho
reason will bo oxplainod when we treat of multiplication.
Ihero are, therefore, several repetitions, which, although
many persons conceive tlioin unnecessary, are not, perhaps,
quite unprofitable. Tho following is free from such an
objection : —
f
Twice 2
» 3
4
6
6
7
8
9
>>
>>
•>
•>
II
II
are 4
— 6
— 8
~ 10
~ 12
— 14
— 18
— 18
8 times 8
4
6
6
7
8
9
II
II
II
II
II
II
9
12
15
18
21
24
27
6 times 7 are 35
8 — 4a
9—45
II
6 times 6
7
8
9
II
II
II
86
42
48
64
4 times 4
6
6
7
8
II
•I
II
II
16
20
24
28
82
86
5 times 6 — 26
II 6—30
7 times 7
II 8
1, 9
— 49
— 66
— 63
8 times 8
., 9
— 64
— 72
9 times 9
81
10 times 8 are 80
II 9 — 90
I, 10 —100
„ 11 —110
10 times 2 are 20
— 80
— 40
— 60
6 — 60
7—70
8
4
6
11 times 2
8
4
6
6
7
8
9
II
II
II
II
II
II
11
22
83
44
55
66
77
88
99
12 times 2 — 24
„ 8 — 86
448
5 60
6  72
7—84
8—96
9 —108
10 —120
11 —182
12 —144
I
II
II
II
II
II
"Ten," or "eleven times," in the above, scarcely requiion
to be committed to memory; since we perceive, that to
multiply a number by 10, we have merely to add a cypher to
tho right hand side of it :— thus, 10 times 8 are 80; and to
multiply it by 11 wo have only to set it down twice :~thus.
11 times 2 rtro 23.
TADLE OF MONEY.
a
2 — 22
8 — 83
i — 44
5 — 55
3 — 66
J — 77
B — 88
) — 99
2 — 24
J — 86
t  48
) 00
5  72
'  84
5 — 96
> —108
> —120
— 182
1 —144
7 requirfl.i
3, that to
cypher to
0', and to
se : — thus,
Tho following tables aro required* for reduction, tho
compound rules, &c., and may bo committed to memory
us convcmonce suggests. ^
TABLE OF MONEY.
A farthing ia the smallest coin generally used in this
country, it is represented bj . . . /""*"'^'
raako 1 halfpenny, i
1 penny, d,
1 shilling, »
Kaitlilngg
o
4 or
48
900
1,008
halt'ponco
24 or
480
504
pence
12
I shillings
240 or 20
252 or I 21
1 pound,
1 guinea.
The symbols of pounds, shillings, and ponce, are placed
over the numbers which express them. Thus 3 14 o"
means, tliree pounds fourteen shillings, and sixpence'.' So'ilic!
times only the symbol for pounds is used, an'd is placed
hofore the whole quantity ; thus, £3 „ 14 „ G. 3 9^ moans
t ireo 8.nlhno8 and mnepence halfpenny. 2s. 6?./. means two
faulhngs and sixpence three farthinf^n, &c
When learning the aI)ove and foflowing tables, the pupil
HhMuld be ri^uired, at hr.st, to commit to memory only those
pnt urns which are over the thick angular linos; thusf in tho
one just given:— 2 farthings make one halfpenny; 2 half
pence one penny; 12 pence one shilling; 20 shillings one
pound; and 21 shillmgs one guinea. a °
Ih h I'oaUy mean the quarter, half, and three quarters
of a penny, d. is used as a symbol, because it is the first
S.
2
5
13
6 make one half Crown.
one Crown.
4 one Mark.
I
4 WEIflHTa.
AVOIRDUPOISE WEiailT.
Its name ia derived from French— and ultimately
from Latin words signify innj " to have weight." It u
used in weighing heavy articles
Drams
10 ... .
2oO ar
7,168
28,672
448 or
1,792
673,4401 85,840
ouncei
16
pound*
28
112 or
2,240
quarteri
4
SymboU
make 1 ounce, oz.
. 1 pound, lb.
. 1 quarter, q.
lhuudrc(l,cwt.
hundred!
lA IK ■^ 80 or I 20 .1 ton. t.
J4 lbs., and m Bomo cases 16 lbs., make 1 stone.
20 stones . . . 1 barrel.
TROY WEIGHT.
It is so called from Troyes, a city in France, where
It was first employed ; it is used in philosophy, in
weighing gold, &c.
Graini » Symboh.
OA •••... grs.
^* • • niftke 1 pennyweight, 'wt.
pennyweights
480 or 20 . . i ounce, o%.
I ounces
6,760^ 240 or I 12 . 1 pound, . lb.
A gram was originally the weight of a grain of corn,
taken from the middle of the ear; a pennyweight, that of
the Sliver penny formerly in use.
APOTHECARIES WEIGHT.
In mixing medicines, apothecaries use Troy weight,
but subdivide it as follows : —
Grains „ . ,
n(\ Symbolt
^^ ' '  ^ » make 1 sc< ipk, ^1
CO or
scruples
3
•
480
24 or
288
drams
8
5,760
90 or
• • • 1 Uium, 6
1 ounce, 5
ounces
12 . 1 pound, lb.
its Carat," which is equal to four grains, is used in
wo.iocMg di«monda. The term carat is alf.o applied in
o^tiin:;nM>; Jie fineness of gold ; the latter, when ju.vf^nti,/
.♦
MBASU1RES.
puro, *i8 said to bo " 24 oarata line." if there are 23 parta
^oid, and one part gomo other material, the mixture is said
to he "23 carats fine ; " if 22 parts out of the 24 are gold,
it is " 22 carats fine," &c. ; — the whole mass is, in all caflos,
supposed to be divided into 24 parts, of which the number
consisting of j»old is specified. Our gold coin is 22 carats
fine; puro gold being very soft would too soon wear out.
The degree of fineness of gold articles is marked upon thera
at I ho Goldsmith's Hall; thus wo generally perceive " 18" on
the cases of gold watches; this indicates that they are " ly
carats fine" — the lowest degree of purity which is stampcfd.
JM.
_ ^ 80
An avoirdupoise ounce . 437^
A Troy pound . . 5,760
An avoirdupoise pound . 7,000
A Troy poimd is equal to 372 9G5 French grammes.
175 Troy pounds are equal to 144 avoirdupoise ;
175 Troy are equal to 192 avoirdupoise ounces.
CLOTH MEASURE.
24
• •
nails
9 or
4
36
16 or
27
12 or
45
20 or
54
24 or
make 1 nail.
quarters
4
8
5
6
1 quarter.
1 yard.
1 Flemish cU
1 English ell.
1 French clI
Linas
12 .
». •
144 or
inches.
12
• •
^432
36 or
198
252
7,920
10,080
8o!640
feet
3
•
2,376
3,024
lOior
21 or
660
840
5,280
6.720
yards
7
95,040
120,960
220 or
280 or
1,760
2,210
7r.().?.20
1167^680
LONG MEASURE.
(It is used to measure Length.)
«. make 1 inch.
perches
40
40
32
on ^
20
or
. 1 foot.
1 yard.
1 English perch
1 Irish perch.
1 English fur long
1 Irish furlong.
a 1 VnliHl. '"!!«
.i^ts^sLjLttrttx ticf 2'9h
8 1 Irish railc.
»?
6
MEASURED
3 iiiclies
3 palms
18 inches
6 f<^.et
6 feet
120 fathoms
3 palm.
1 span.
1 cubit
1 pace.
1 fathom.
1 cable's length.
rnHpr rii'?/ ^ . . ^ ^"^^ a^e equal to 14 Endisli
miiec. Ihe Pans foot is eaual tn 197Q0 v^ vl ."&"^^*
the Roman foot to 11604 liTih^ ^ u ^""^^'"^ '"^^««J
luuL 10 ii OU4, and the French metre to 39383.
, MEASURE OP SURFACES
foot, a surface one foot long and one foot m^L', L.^
Square inches
144
1,996
3&,304
63,604
^,668,160
9,640,160
6,272,640
10,160,640
ncake 1 «q. foot.
1 square yard.
10,890
17,640
{43,560
70,560
1 «q. En,
1 sq.Iris
porcli.'
perah.
4,014,489,600 27,878,400
6,602,809.600i25. 158,400
1 sq. Eng. rood. '
• «q. Irish rood.
1 statute acre.
1 plantation acre.
3,097,600 102,400
5,017,60oll02,400
1 sq. En,
1 sq. Iris
mile,
mile.
rp, „ ' ■ • "'" 1 sq. irisn mile.
crXfaff 'aoTmo^ 1'or" ^"ejarda, and the Irish.
12lVre Irish aero, ' l"" «1™'« Kngibi are equal to
anl'The^lg' IKoS''^o'^T^'fJ'"'" ^", ^^^
en„o] f,. 101 r.:!.'. ' ,' "^"^^ J^^nglish square milfis a^a
MKASURES.
MEASURE OP SOLIDS.
^ The teacher will explain that a cube is a solid having
six equal square surfaces; and will illustrate this by
models or examples — the more familiar the better. A
cubic inch is a solid, each of whoso a^x sides or faces is
a sqitare inch ; a cubic foot a solid cooh of whose osi
sides is a square fool ^ &c.
Cubic inches
I'^^^S . . , . . niftka 1 cub"? fot^vV
I cubic feet
27 . . . icubu5r»J4\
WINE MEASURE.
Gills or naggina
8 or
32
320
576
1,344
2,016
2,688
pints
')
^ m %
•
•
quarts
8 or
4
•
•
3
•
gallon
80
40 or
10
«
•
144
72
18
•
«
836
168
42
.
•
504
252
63
•
•
672
336
84
•
hogshe
•
Elds
1,008
2,016
504
1,008
126 or
252
2
•
4 or
xnai^^ i <«i».*.
1 gaJoiX
1 anker.
1 runlet.
1 tierce.
1 hogsheaa
1 punclieon
pipes
2
4,032
8,064
in some places a gill is equal to half a pint.
Foreign wines, &c., are often sold by measures differing
from the above.
1 pipe or butt
1 tun.
ALE MEASURE.
Gallons
8
16 or
firkins
2
82
4 or
6
8
12
kilderkins
2
48
64
96
8 or
4 or
6 or
barrels
u
make 1 firkin.
1 kilderkin.
1 barrel.
1 hogshead.
1 puncheon.
1 butt.
MEASURES.
BEER MEASURE.
make 1 firkin.
1 kilderkin
1 barrel.
1 hogshead.
1 puncheon.
1 butt.
Pints
4 or
8
16
64
152
256
576
DRY MEASURE.
(It is used for wheat, and other dry goods.)
quarts
2
512
2,048
2,500
4 or
8
32
96
12V.
288
2uG
1,021
1,280
pottles
2
5,120 2,560
4 or
16
48
64
144
128
gallons
2
512
640
1,280
8 or
24
32
72
64
256
320
640
pecks
4
12or
16or
36 or
32
128
160
bushels
3
4
9
320 80
8 or
32
40
coombs
2
make 1 pottle.
1 gallon.
1 peck.
1 bushel.
1 sack.
1 coomb.
1 vat.
8 or
lOor
20
quarters
4
5
1 quarter.
1 chaldron
1 wey.
weys
10 or I 2 Hast.
riurds
60
TIME.
MEASURE OE TIME.
or
3600,
216,000
5,184,000
t 0,288,000
. 15,152,000
1,892,160,000
1,897,344,000
1,892,160,000
seconds
60
3600 or
86,400
604,800
2,419,200
31,536,000
31,622,400
31,636,000
minutes
60
1,440 or
10,080
40,320
625,000
627,040
525,600
hours
24
Ifi'^ 01
672 01
8,700 or
8,784 01
8,760
Jays
7
28
305
366
365 or
SytiaoU
make 1 second "
1 minute «
1 hour h.
1 day d
1 week w.
1 lunar month.
1 common year
1 leap year.
calendar mon.")
12 I ,
lunar months f * y^^^'
13 J
The following Avill exemplify the use of the above symbols : — 
The solar year consists of 365 d. 5 h. 48' 45" 30'": read » three
hundred and sixtyfive days, five hours, fortyeight minuteB.
fortyfive seconds, and thirty third;?.
The number of days in each of the twelve calendar months
will be easily remembered by means of the well known lines,
"Thirty days hath September,
April, June, and November,
February twentyeight alono
And all the rest t)iirty»ne."
The follomng table vrill enable us to find how many days
there are from any day in one month to any day in another.
From any Day in j
T.
M
>•
Cl
>•
o
H
i
Jan.
Feb.
Mar
April
May
June
July
Aug.
Sept.
Oct.
Nov
1
Dec
Jan.
36.5
334
306
276
245
214
184
163
122
92
61
31
Feb.
31
36.')
337
306
270
304
246
215
184
153
123
92
62
90
Mar.
59
2S
366
334
273
243
212
131
151
120
April
90
69
31
365
.335
304
274
243
212
182
151
121
May
120
89
61
30
365
334
304
273
242
212
181
161
June
July
151
181
120
92
61
31
365
336
304
273
243
212
182
150
122
91
61
92
30
6)
366
334
303
273
242
212
243
Aug.
212
181
153
122
31
365
334
304
273
Sei)t.
243
212
184
153
123
92
62
31
365
335
304
274
Oct.
273
242
214
183
153
122
92
61
30
366
334
304
335
Nov.
304
273
245
214
1.4
153
123
92
61
31
365
Dec.
33^1
303 275
211
214
183
163
122
91
61
30
366
I
10
TIME.
iH plucod, and at the samJSf.' i *^^\^«^d of which March
ihi left hand sSe of wWch ? On nT^ ^^' horizontal row at
intersection the numW o]] • t '"' "^^ P"""^^^^ ^'^ ^^^^r
tervene between ti el^fceenth 'oT M T clays therefore, in
October. But thrfourth nf o!. k '"' ' ^?^ *^^ fifteenth of
than the fifteenth Zv?, 5/^ *"^T '" ^^^^«" d^y« earl'er
obtjdn 20i'S::r!^^^:^ ^^'^^^^^ " ^ ^U, and
^f^^^^^^ the
bofore in tlie table, wo find tint Tin^ • ? Looking as
th., third of J„„„,,y and tt th rd "ffc"t7rt^'"'^''"
tucntli IS sixteen days later thnn *!,„.;■ J' "'f' ""' "'"e
«nd obtaii, 136, the Jiit, required ■*' ™ ''^<' ^« '» ^^^
«Ud one to the 130, aXl'sV tuldtZlt^'r "^ "'"'"^
'w this bclZTnZl^lTtTl ^" "■«J"«tand
tbe Julian Calenda,;S !"«',,» T '" '."'""^ """
fourth year to the mojrirSarf tftM J P • ''^
Gregory, t.ro..d, this: „..dai™r«,s^;:;::;„„^r
TIME
11
to the Julian style, would have been the 5th of October
1582, should be considered as the 15th ; and to preveni'
the recurrence of such a mistake, he desired that, in
place of the last year of every century being, as hitherto,
a leap year, only the last year of every fourth century
should be deemed such.
The " New Style," as it is called, was not introduced
into England until 1752, when the error had become
eleven days. The Gregorian Calendar itself is slightly
inaccurate.
To find if any given year be a leap year. Tf net the
last year of a century :
Rule. — Divide the number which represents the
given year by 4, and if there be no remainder, it is a
leap year. If there be a remainder, it expresses how
long the given year is after the preceding leap year.
Example 1.— 1840 waa a leap year, because 1840 divided
by 4 leaves no remainder.
Example 2. — 1722 was the second year after a leap year,
because 1722 divided by 4 leaves 2 as remainder.
If the given year be the last of a century :.
ivULE.— Divide the number expressing the centuries
by 4, and if there be no remainder, the given one is a leap
year ; if there be a remainder, it indicates the number
of centuries between the given and preceding last year
of a century which was a leap year.
Example 1. — 1600 was a leap year, because 10, being
divided by 4, leaves nothing.
Example 2.— 1800 was two centuries after that last year
of a century which was a leap year, because, divided by 4,
it leaves 2. •
DIVISrW OF THE CIRCLE.
Thirds
60
8600 or
216,000
77,760,000
seconds
60
8,600 or
1,296,000
minutca
60
make 1 second "
1 minute '
1 degree °
I degrees
360 1 circumference.
. Ev«»vy circle is supposed to be divided into the same
P'^.'^^r of degrees, miautos, &c. ; the greater or less, there
12
DEFlNrnONf?.
fore, the circle, the greater or less each of these will be. The
following will exemplify the applications of the symbolej : —
00° 6' 4" 6'" ; which means sixty degrees, five minutes, four
sccondj, and six thirds.
DEFINITIONS
1. Arithmetic may be considered either as a science
or as an art. As a science, it teaches the properties of
numbers ; as an art, it enables us to apply this know
ledge to practical purposes ; the former may be called
theore(tical, the latter practical arithmetic.
' 2. J. Unit^ or as it is also called, Unity ^ is one of tho
indivdduals nnder consideration, and may include many
units of another kind or denomination ; thus a unit of
the order called " tens" consists of ten simple units. Or
it may consist of one or more parts of a unit of a higher
denomination ; thus five units of the order of " tens" are
five parts of one of the denomination called " hundreds ;"
three units of the denomination called " tenths" are
three parts of a unit, which wo shall presently term *the
" unit of comparison."
3. Numler is constituted of two or more units ;
strictly speaking, therefore, unity itself cannot be con
sidered as a number.
4. Abstract Nwmhers are those the properties of
which are contemplated without reference to their appli
cation to any particular purpose — as five, seven, &c. ;
abstraction l)eii^ a process of the mind, by which it sepa
rate^/ considers those qualities which cannot in reality
exist by themselves ; thus, for example, when we attend
only to the length of anything, we are said to abstract
from its breadth, thickness, colour, &c., although these
are necessarily found associated with it. There is nothing
inaccurate in this abstraction, since, although length
cannot exist without breadth, thickness, &c., it has pro
perties independent of them. In the samo way, five, seven,
&c., can be considered only by an abstraction of the
wind, as not applied to indicate soiae particuiar things.
5. Ajjplicate Numbers are exactly the reverse of
DICFINITIONS.
13
(rill be. The
! symbolej : —
aiuutes, four
IS a science
roperties of
this know
y be called
i one of tlio
elude many
i a unit of
e units. Or
of a higher
" tens" are
hundreds ;"
;enth8" are
ly term 'the
lore units ;
lot be con
aperties of
their appli
even, &c. ;
tich it sepa
t in reality
Q we attend
to abstract
ough these
e is nothing
ugh length
it has pro
, five, seven,
tion of the
xliii' things,
reverse of
abstract, being applied to indicate particular objects —
as five men, six houses.
6, The Unit of Comparison. In every number
there is some unit or individual which is used as a
standard : this we shall henceforward call the " unit
of comparison." It is by no means necessary that it
should always be the same ; for at one time we may
speak of four objects of one species, at another of four
objects of another species, at a third, of four dozen, or
four scores of objects ; in all these cases four is the
number contemplated, though in each of them the idea
conveyed to the mind is different — this difference arising
from the different standard of comparison, or unity
assumed. In the first case, the " unit of comparison"
was a single object ; in the second, it was also a single
object, but not of the same kind ; in the third, it became
a dozen ; and in the fourth, a score of objects. Increas
ing the '' unit of comparison" evidently increases the
(Quantity indicated by a given number ; while decreas
ing it has a contrary effect. It will be necessary to
bear all tliis carefully in mind.
7. Odd Numbers. One, and every succeeding alter
nate number, are termed odd ; thus, three, five, seven, &c.
S. Evm Numbers. Two, and evcy succeeding alter
nate number, are said to be even ; thus, four, six, eight,
&c. It is scarcely necessary to remark, that after taking
away the odd numbers, all tliose which remain are even°
and after taking away the even, all those which remain
arc odd.
We shall introduce many other definitions when treat
ing of those matters to which they felate. A clear
idea of what is proposed for consideration is of tho
greatest importance; this must be derived from tho
definition by which it is explained.
Since nothing assists both the understanding and the
memory more than accurately dividing the subject of
instruction, we shall take this opportunity of remarking
to both teacher and pupil, that we attach much impor
tance to the divisions which in future shall actually be
madr;, or shall be implied by the order in which tho
different heads will be examined.
b2
M
I t
SECTION I.
ON NOTATION AND NUMERATION.
1. To avail ourselves of the properties of numbers,
we must be able both to form an idea of them ouiselvcs,
and to convey this idea to others by spoken and by written
language ; — that is, by the voice, and by characters.
The expression of number by characters, is called
notation, the reading of these, numeration. Notation,
therefore, and numeration, bear the same relation to
each other as loriting and reading, and though often
confounded, tliey are in reality perfectly distinct.
2. It is obvious that, for the purposes of Arithmetic,
we require the power of designating all possible num
bers ; it is equally obvious that we cannot give a dif
ferent name or character to each, as their variety is
boundless. Wo must, therefore, by some means or
another, make a limited system of words and signs
Buffice to express an unlimited amount of numerical
quantities : — ^with what beautiful simplicity and clear
ness this is effected, we shall better understand presently.
3. Two modes of attaining such an object present
themselves ; the one, that of comhining words or cha
racters already in use, to indicate new quantities ; the
other, that of representing a variety of different quan
tities by a single word or character, the danger of
mistake at the same time being prevented. The Romans
Bimplified their system of notation by adopting the prin
ciple of combination ; but the still greater perfection of
ours is due also to the expression of many numbers by
the same character.
4. It will be useful, and not at all difficult, to explain
to the pupil the mode by which, as we may suppose, an
idea of considerable numbers was originally acquued,
and of which, mdeed, although unconsciously, we still
avail omselves ; we shall see, at the same time, how
methods of simplifying both numeration and notation
Were naturally suggested.
NOTATION AND NUMERATION.
15
of numbers,
im ourselves,
id by written
aracters.
rs, is called
. Notation,
relation to
hough often
;inct.
Arithmetic,
)ssible num
I give a dif
ir variety is
e means or
3 and signs
if numerical
^ and clear
id presently,
ject present
)rds or cha
mtities ; the
Ferent quan
! danger of
rhe Romans
ing the prin
lerfection of
numbers by
t, to explain
suppose, an
ly acquu'ed,
isly, we still
3 time, how
.nd notation
I^.' lis suppose no system of numbors to bo as yo.i con
^(ruered and that a licap, for oxan.ph,, of pebbles, i.s
p Mcod before us that wo may discover their amount.
It this IS con.sulerablo, we cnimot ascertain it l)y look
ing at tlicui all together, nor even by separatdy in
specting them; we must, therefore, have recourse t(»
that contnvnuce which the mind always uses when it
desires to grasp what, taken as a whole, is too great for
IS powers. It we exan.inc an extensive landscape, as
the eye cannot take it all in at out view, we look sue
cessively at its different portions, and form our iud.^
n.ont upon them in detail. We must act similarly with
retcrerice to large nunibers ; since we cannot compre
Jiend^thcm at a single glance, we must divide them into
a suHiciout number of parts, and, examining these in
succession, acquire an indirect, but accuratS idea of
he entire. This process becomes by habit so rapid,
that It seems, it carelessly observed, but one act, thouh
It is made up of nuiny : it is indispensable, whenc^cr wo
clfsire to have a clmr idea of nunibeis— which is not
Iiowcver, every time they are mentioned. '
5. Had we, then, to form for oursolVes a numerical
sys eni, we would naturally divide the individuals to be
reckoned into equal groups, each group consisting of
S;mio number quite within the limit of our comprJlien
rum ; it the groups were few, our object would be attained
vithout any further effort, .incc .ve should have acquired
•urate kiiow edge of the number of groups, aiid of
•n.l^er of individuals in each group, and therefore
i ik.'tory, although mduect estimate of the whole
>.e^^ught to remark, that different persons have
ery different Innits to their perfect comprehension of
number; the mteliigent can conceive with ease a com
incapable of forn)mg an idea of one that is extremelv
flmali. •'
6. Let us call the nmnhr of individuals that we choose
cons itute a group, the rafio ; it is evident that the
larger the ratio, the smaller the number of groups ano
the smaller the ratio <!. Uyo^^ fi.p ,,,,.i fe^""l''^> 'in^
&ut the smaller the number of groups the better.
16
NOTAT[()?J AND NUMKRATION.
7. If the groups into which wo havo divided tho
objects to bo rockoueJ exceed in amount that number
of whicli we havo a i)orf(!et idea, wo must continue tho
process, and considorinj,' tho groups tlieniHolvos a.s indi
viduals, must form with them new groups of a higher
order. Wc must thus proceed until tho number of^our
highest group is sutFiciently .« ' ill.
8. Tho raiio used for groups of the second and higher
orders, would naturally, but not necessarily, be the samo
as that adopted for the lowest ; that is, if seven indi
viduals constitute a group of tlie first order, we would
probably make seven groups of the first order constitute
a group of the second also ; and so on.
y. It might, and very likely would happen, that wo
should not have so many objects as would exactly form
a certain* number of groups of the highest order —
eomo of the next lower might be left. The same might
occur in forming one or moi'e of the other groups. Wo
might, for example, in reckoning a heap of pebbles,
have two groups of the fourth order, three of the third,
none of the second, five of the first, and seven indi
viduals or "units of comparison."
10. If wo had made each of the first order of groups
consist of ten pebbles, and each of the second oider
consist of ton of the first, oacli group of the third of tm
of the second, and so on with the rest, we had selected
the deciiiuU system, or tliat which is not only used at
present, but which was adopted by tho llcjbrew.s, Greeks,
Komans, &g. It is remarkable that the language of
every civilized nation gives names to the ' dilferent
groups of this, but not to those of any other numerical
system ; its very general diffusion, even among rude
Hnd barbarous people, has most probably arisen from
the habit of counting on the fingers, whidi is not
altogether abandoned, even by us.
11. It was not indispensable that we sliould havo
ased the same ratio for the groups of all tho diiFereut
orders ; we might, for example, have made four pebbles
form a group of the first order, t^velve groups of the
first order a group of the second, and twen.ty grouT)a
of the second a group of the third order :— iu such a
i
(lividcd tho
; that number
b continue tho
wives aH incli
is of a hii^lior
lumber of our
nd and higher
r, be the same
if seven indi
ler, we would
der constitute
ppcn, that wo
exactly form
jhcst order — ■
10 same mip;lit
groups. \Vo
p of pebbles,
1 of the third,
i seven indi
dor of groups
second ordor
e third of ton
had selected
only used at
rews, Greeks,
language of
tho diiibrent
ler numerical
among rudo
arisen from
vhich is not
should havo
tho diiForent
four pebbles
•roups of the
vonty o'voiiT)a
:— iu such a
NOTA'lloN AND NUATERATION. I7
case we had adopted a system oxacllj Hko that to \m
l.i. things make a group „f the order pma' twolvo
pcnro a g,H,up <,f the order di/fin^s, twenty shillms^
group of tho o.lor rour.ls. While ii urns "L ad LS
that the use of tho su>ne system for anplicato ■ s fV, r
p^ ^:"e "';;f 'vv^'' sutiy si,..piity ^r :;is;."ti^
?iT •''■' H' ''''y ^'''^'^"^ hereafter, a dance
^}t the tab OS g,v.m already, and those set down in t^^o' tt
ag of exciyjugo, will .how that a great vanoZ^X^
have actu;illy been constiuctcd. «>> stems
12. When wo use the same ra/io for tho groups of all
h.. orders, we term it a cnmuon ratio. The,^ aX' s
I'tvo boon nr> particular reason why /...slumid So Lee.
seloctod as a ' connuon ratio" i„ the systeuM/f numbe'
.a.,1 od, by the mode of counting on the fimrors • a„,l
that It is neither so low as unnecessarily Si .Ve
'0 number 01 orders of groups, nor so higlf afto oxc^^i
J^o ...jceptmn of any one ibr whom the sys;;^^'
. ■'"!; ^^;^y«^^^'" i'l ^^hi«li ten is the "common ntio^'
r.n — ouis IS, taoroforo, a " docimal .'system" of numbe.s
It the common ratio were sixty, it would b^ a y ' . • "
^^^: :;;;'7.^"^^ i^ony usod,;ndt^m
i e ft. 'L T P^^'^'^ived by the tables already
g.vcn toi the measurement of arcs and anlos and of
time. A ^n,na,y .system wOuld have five ^ r k s'  com
mon ratio  a cluoaecnal, twelve ; a vl^.i^^^l: tv.^nt^;
14. A little reflection will show that it was useloss
to give diflorent ikumos and chT.,pfp,.« f/ "■''^^loss
except to those which a "lo^'h ^ " hnt^'ILw^'l"
tutos the lowest g.oup, and to th #.:; 1 ^ilf
ss.;.d^r:v^:^ ''''''' nun;boi.iast'c:;:^;:
oviaent4ib:u;:;;^h;;;r^t:;r^^^^^
18
NOTATION AND NUMERATION.
Nnnttt,
Chitntetvt.
One
1
Two
9
Til 100
.<}
l''oiir
4
Kivo
A
Six .
6
Se'en
7
Kiiht .
8
Nu.o
SI
Ten
10
Hundred
luo
'J'lioiisund
1,000
'J'eii tliotiKn
nd"
lo.wm
lliiiuliud til
Dusa
nd ico.ooo
Million .
.
1,000,000
is just what wo liiivo dnno in our numnrioal py.stoni,
excofit tliat we have rormcd tho nnnioH of koiuo of tlio
groups )»y cmubination (»f tlioso nlrcndy used ; thus,
*' tens of tliouvinTi(i.M," the grou]) n;'xt IihtIkm' th.in lliou
sandn, is designated by a conibinntioii of words ahcady
ajtpliod to express other groups — whieli tends yet further
to siniplilication.
15, ARADIC SySTEJM OF NOTATION :
UdUs ol uonijiai'lson,
First grotip, or units of tho second order,
Seooiiii ({roup, or iiiiitti of the third ordt.r,
Tliinl Kronp, or units of tho fourth order,
Fourth groii]), or units of the llfth order,
l''ilih f;r()ii), or units of the sixth order,
Sixth gronii, or units of tho .seventh order,
Ifi. The characters whicli express tlio nine first nuTu
bers are tho only ones used ; they are called (/i.gi/s^ JVoin
tho custom of counting thoni on the lingers, already
noticed — " digitus" meaning in Latin a finger ; Ihay aie
also called significant figures, to distinguish them tV.,m
the cypher, or 0, which is used merely to give the digits
their proper 'position with ret'crcncc to tht; dcciinal point.
The pupil will distinctly remember that the place where
the "• units of comparison" are to be found is that imme
diately to the left hand of this point, which, if n(»t ex
pressed, is supposed to stand to the right hand i>id(.; of
all tho digits — thus, in 4(JS7G tho 8 expresses " units
of comparison," being to the left of the decimal point ;
in 40 the 9 expresses " units of comparison,'"' the deci
mal point being understood to tlie riirlit of it.
17. We find by the table just given, that after the
nine first numbers, the same digit is constantly repeaieil,
its position with reference to tho diciaal point being,
however, changed : — that is, to indicate each succeeding
group it is moved, by means of a cyplier, one place
farther to the loft. Any uf the dibits may be uaed to
N.
fiorionl pystoni,
of 8omo of (lio
y usod ; thus,
rlier tlinn tliou
wotds nlrcjuiy
!U(Is yot further
Chantetvt.
1
a
4
A
6
7
8
SI
10
10(1
I,(J()()
10,(HI0
NOTATION AND Nt/MERATIOrf
19
!<1
ind
niKnnd
d thousand l(:0,(il)(i
l,()00,f)(;o
nine fiiHt miTii
iod ili.gih^ JVoiii
iiigor.s, already
iigcr ; Ihay are
ish thciiu fr.,"in
give tlie di^i<s
! decimal p. int.
he place where
d \H that imtue
ieh, if nut ex
t hand side of
presses " uuitH
decimal point ;
son,'' the deci
■ it.
that after tlse
antly repeated,
d i)oint being,
ach succeeding'
lier, one place
ay be used to
flxpreHs its respectiv(3 n.iniber „f any of the .rroups •.
thus S would be eight u ...j,, ./.nnpari^onTH^
oiht groups ot tho first ord<n, or eidit " tens" of
Mmpic units ; 800, eight groups of tho .second, or' unit^
f tho h.rd order ; and so on. Wo might use any of
tho digits With tho different groups ; thus, for examDio
7 ;;;; r^'n^^v'^'^ "^i"^^ 3?o;those'of;he " S;
•Z'to r *'^:: ^^^1^''" •?^'*^ J™ i'l f>ill would be 5000,
3 0, 70 8, or for brevity sake, 5378for wo never nm
tho cypher when we can supply its place by a si^^nificant
ftguro, and it is evident that in 5378 t!ie 378 ko p^^^^^^^
T) tour p.aees from th. decimal point (understood), fult aa
wo as cyphers would have done ; a\so the 78 k^ee'pT ho
3 Y the third, and the 8 keeps the 7 in tho second placo.
lb. It IS important to remember that each di^rit hag
two values, au absoUta and a relative; tho absolute
un " m v' h ""'"i '' ""^^^^ '^^'^'^'^^ whatever tte
units may bo, and is unchangeable: thus 6 alwavs
I'/oans SIX, sometimes, indeed, Sx tens, at otlu^r tSa
Hix hundred, &c. The relative value depends on the
o dcr of units mdicated, and on the nature if tho "unit
ot comparison." *
_ la. What has been said on this very important snli
l;;ct, IS intended piincipally for the teLheir both t
o.dmary amoun of industry and intelligence wil be
a ci ld,_ paitieulariy it each point is illustrated by an
appropriate example ; the pupil may be made f\ f in
.stance, to arrange a numbe/of pebbles in groups son "
ti.nes of one, sometimes of another, and soiZinies of
S'od f 1 ;n7 • 1 1 '°'"P^'''^'' ^^'^"^o^ occasionally
,, Changed tiom individuals, suppose to tens, or hundreds op
\ scores, or dozens, &c. Indeed the pupils J fbe well
; oquamted witli these introductory ma^tters otherw^o
., verj acnnite ideas of many th n'>s thev will be poIl/i
inijc£ii.y to understand. A.n" t"0'ihia Un •f.ixr i i ,^
toaoho. at thi. „.,„,, ,,ni f,o we,f;o,«i;n;'ttt
(i
■
111
20
NOTATION AND NrMliRA'I'IOV.
and rapidUj^ with Aviiicli tlic sdiolar will cafterwardg
adviincjo ; to be assured of this, ho has only to recol
lect that most of his future reasonings will be derived
from, and his explanations grounded "on the very prin
ciples we have endeavoured to unfold. It may be taken
as an important truth, that what a child learns without
undcratanding, he will acquire with disgust, and will
uoon cease to remember ; for it is with children as with
persons of more advanced years, when wo appeal suc
cessfully to their understanding, the pride and pleasure
they feel hi the attainment of knowledge, cause the
labour and the weariness which it costs to be under
valued, or forgotten.
20. Pebbles will answer well for examples ; indeed,
their use in computing has given rise to the term calcu
lation., " calculus" being, in Latin, a pebble : but while
the teacher illustrates what he says by groups of par
ticular objects, he must take care to notice that hi^^
remarks would be equally true of any others. He must
also point out the difference between a group and its
e(juivak'nt unit, which, from their perfect equality, are
generally confounded. Thus ho may show, that a penny,
while C(iual to, is not identical with four farthings. This
seemingly unimportant remark will be better appre
ciated hereafter ; at the same time, without inaccuracy
of result, we may, if we please, consider any group
dther as a unit of the order to wliieh it 1)clongs, or so
many of the next lower as are equivalent.
21. lloman Notaiion. — Our ordinary numerical cha
racters have not been always, nor every where used tc
express numbers ; the letters of the alphabet naturally
piesonted themselves for the puipose, as being already
familiar, and, accordingly, were very generally adopted—
for example, by the Hebrews, Greeks, llomans, &c.,
each, of course, using their own alphabet. The pupil
should be acquainted with the lloman notation on
account of its beautiful simplicity, and its being still
employed in inscriptions, &c. : it is found in the follow
ing table : —
ill afterwardg
only to rccol'
dll be derived
:lio very prin
may bo taken
learns without
;nst, and will
ildren as with
e appeal suc
and pleasure
50, cause the
to be under
ples ; indeed,
le term calmt
le : but while
roups of par
tice that \m
rs. He must
;roup and its
equality, are
that a penny,
'things. This
better appre
Lit inaccuracy
r any group
)elongs, or so
imerical cha
vhere used tc
.bet naturally
being already
lly adojited — ■
iomans, &c.,
u The pupil
notation on
ts being still
in the follow
NOTATIOi'T AND NUMERATION.
ROMAN NOTATION.
31
Characters.
I.
n.
in. .
Anticipated change IIII, or IV.
OJiauge . . V. .
VI. .
VII. .
VIII. .
Auticipaletl change IX.
Change
X.
XI.
XII.
XIII.
xtv.
XV.
XVI.
XVII.
XVIII
xrx.
XX.
A uticipated change X [,
Change . . L.
. . T'X., &c.
Anticipated change XO.
Change . . C.
, . ". CO., &c.
Anticipated change CD.
Cliango . . D. orr>.
Anticipated change CM
Ciiauge , ' . U. or CIq
y. or 1,30
&c
JSiimhers Exprrsstd.
. One.
. Two.
. TJiree
. Four.
. Five.
. Six.
. Seven.
. Eight.
. Nine.
. Ten.
. Eleven.
. Twelve.
. Thirteen.
. Fourteen.
Fifteen.
Sixteen.
Seventeen.
Eighteen.
Nineteen.
Twenty.
Thirty, &c.
Forty.
Fifty.
Sixty, &c.
Ninety.
One hundred.
Two hundred, kc
Four hundred.
Five hundred, &c
Nine hundred.
One thousand, &c.
Five thousand, &c.
X. or CCIoo • Ten thousand, &o.
laop. . Fifty thousand, &cj.
'^''CiOOO. • One liundred thous.and, &c
. 2,2. Thus we find that the liomuns used vnrv fe^7
eiiaractersfower, iridoed, than we do, althouWl our
sjstoMi ,s stdl more snaplo and eiToctive, from our ap,>K
ing.tli. prmc.pl. of "position," unkuo.vn to them ^
llicy expressed all numbers by the followiiv. symbol.
or combmatiou.3 of them : I. Y. X. L. (J. D. ov U T"
orCLo.^ In cojistrueting their system, they evidently
had a (pmary m view ; that is, as we have ^rid, an. I
wlucii nve would be the common rallo ; for we find that
luey ciians'tiu th
heir eharactur, not only at ten, toij t
iine»
i i
in.
22
WOTAT ^,^ Aww wwMEKATION.
ten, &c., but also at five, ten times five, &c. : — a purely
decimal system would suggest a change only at ten, ten
times ten, &c. ; a purely quinary, only at five, five times
five, &c. As far as notation was concerned, what they
adopted was neither a decimal nor a quinary system,
nor even a combination of both ; they appear to have
supposed two primary groups, one of five, the other of
ten " units of comparison ; " and to have formed all the
other groups from these, by using ten as the common
ratio of each resulting series.
23. They anticipated a change of character; one
unit before it would naturally occur — that is, not one
" unit of comparison," but one of the units under consi
deration. In this point of view, four is one unit before
five ; forty, one unit before fifty— tens being now the
units under consideration ; four hundred, one unit before
five hundred— hundieds having become the units con
templated.
24. When a lower character is placed before a
higher its value is to be subtracted from, when placed
after it, to be added to the value of the higher ; thus,
IV. means one less than five, or four ; VI., one more
than five, or six.
2b. To express a number by the Roman method of
notation : —
Rule.— Find the highest number within the given
one, that is expressed by a single character, or the
" anticipation " of one [21] ; set down that character,
or anticipation— as the case may be, and take its value
from the given number. Find what highest number
less than the remainder is expressed by a single charac
ter, or " anticipation ; " put that character or " anticipa •
tion "^ to the right hand of what is already wi'itten, and
take its value from the last remainder :' proceed thus
until nothing is left.
Example.— Set down the present year, eighteen h'mdred
and fortyfour, in Roman characters. One thousand, ex
pressed by M., is the highest number within the giynn one,
indicated by one character, or by an anticipation; we pu k down
and take one thousand from the given number, which loaves
^S
r.
ic. : — a purely
tily at ten, ten
EJve, five times
ed, what they
linary system,
ppear to have
, the other of
formed all the
LS the common
laracter ; one
it is, not one
3 under consi
ne unit before
eing now the
ne unit before
he units con
jed before a
when placed
ligher ; thus,
I., one more
n method of
lin the given
icter, or the
at character,
take its value
^hest number
lingle charac
or " anticipa •
written, and
proceed thus
teen h'lndred
thousap.d, ex
the givon one,
; we pu k down
which loaves
NOTATIOM AND NCTJIERATION.
23
'IS'
eight hundred and fortyfour. Five hundro „ tho
highest number within tho last remainder (e /./^ fu.r,dred
and ioryfour) expressed by one character, or jin "antici
pation ;■' we set down D to the rin;ht liand of M
JNID,
and take its value from eight hundred and fortyfour, which
leaves three hundred and fortyfour. In this^the hiohost
number expressed by a single character, or an "anticipa
tion, ' IS one hundred, indicated by C ; which we set down ;
and tor the same reason two other Cs
Tw , MDCCC.
This leaves only fortyfour, tho highest number withic
which, expressed by a single character, or an "anticipation ''
IS torty, AL— an anticipation ; we set this down alsof
_ MDCCCXL.
Four expressed by IV., still remains; which, being als«
added, the whole is as follows:— « uj, ms*
MDCCCXLTV.
26 Posiiion.—The samp .jharacter may have dif
terent values, according to tie place it holds with refer
ence to the decimal point, or, perhaps, more strictly,
osiiion comparison." This is the principle of
27. The places occupied by the units of the different
orders, according to the Arabic, or ordinary notation
[lo] , may be described as follows :units of comparison,
one place to the left of the decimal point, expressed
01 understood ; tens, two places ; hundreds, three places
tl K 1 F""^'^ '^'''"^•^ ^' "^^'^^ '^'^ ^^™ili^r with these
as to be able, at once, to name the « place" of any order
of units, or the " units" of any place ^
28 When therefore, we are deshed to write anv
number, we have merely to put down the digits expres
sing the amounts of the different units in their prler
places, according to the order to which each belongs,
it, in the given number, there is any order of which
domi u the place belonging to it ; the object of which
2r^ ^7 'I::;, "Snifica. t figures in^ thpir own posi
mns. A cypher produces no effect wh
between significant figu^^s and the
0536, 6360, and 536
on it is not
decimal point; thus
would mean the same thing — the
24
NOTATION AND NUMERATION.
second s, however, the correct form. 536 and 5360 dre
different ; in the latter case the cypher affects the value
because it alters the position of the digits. '
Example —Let it be re uired to set down si^ hundred
h' ; ' Sr "^*^,"«' ^^2 : without the cypher, the six would
29. In numerating, we begin with the digits of the
highest order and proceed downwards, stating the num
ber which belongs to each order, fe nuiu
clJhV'tf'^''^'' notation and numeration, it is usual to
divide the places occupied by the different ordei of
units into periods ; for a certain distance the Englisn and
1 rench methods of division agree ; the English bHlion
IS, however, a thousand times greater than the French
Ihis discrepancy is not of much importance, since we
aie rarely obkged to use so high a number,we shall
prefer the IWh method. tS give some idea of the
amount of a billion, it is only necessary to remark that
according to the English Method ot' notation 'there
lias not been one billion of seconds sin e the birth of
thrist. Indeed, to reckon even a million, counting on
an average^ three per second for eight hours a day,
would require nearly 12 days. The following are the
two methods. °
\*
Jill
ENGLISH METHOD.
Twin nnn B^^ions. Millions.
000000 000000 000000
Units.
000000
Billions.
Quiidivila. 'J'ens. Unita.
FRENCH METHOD.
Millions. Thousands. Units.
Himd. .ens. Units. H„nd. Ten.. Unita. Hnnd, Ton.. Unit*
000 000 000
so Use of Periods.— Let it be required to read off
the following number, 576934. We put the first point
to the left of the hundreds' place, and find that there are
exactly two periods— 576,934 ; this does not always
occur, as the higlicst period is often imperfect, consisting
oniy 01 one or two digits. Dividing "the number thus
NOTATION AND NUMERATION.
and 5360 jlre
cts the value,
1 six hundred
the two in the
^her between
the six wouhi
mean not six
digits of the
ng the num
t is usual to
it ordei of
Englisii and
iglish billion
the French,
ee, since we
p, — ^we shall
idea of the
•emark, that
ation, there
he birth of
counting on
ours a (lay,
ing are the
Units.
000000
Units.
nnd. Tpi)«. Unit*
to read off
i first point
it there are
not always
, consisting
imber thus
2b
into part,^, shows at once that 5 is in the third place of
the second period, and of course in the sixth place to
the 4eft hand of the decimal point (understood) ; and,
therefore, that it expresses hundreds of thousands. The
7 being in the fifch place, indicates tons of tliousands ;
the tJ in the fourth, thousands ; the 9 in the third, hun
dreds ; the 3 in the second, tens ; and the 4 in the first,
units (of " comparison ") . The whole, therefore, is five
hundreds of thousands, seven tens of thousands, six
thousands, nine hundreds, three tens, and four units,—
or more briefly, five hundred and seventysix thousand,
nine hundred and thirtyfour,
31. To prevent the separating point, or that which
divides into periods, from being mistaken for the decimal
point, the former should be a comma (,)— the latter a
tuli stop (•) Without this distinction, two numbers
Y.o^'i^f'' ^?^ different might be confounded : thus,
498 /63 and 498,763,one of which is a thousand
times greater than the other. After a while, we may
dispense witli the separating point, thouoh it is conve
nient to use It with considerable number, as they are
tiien read with greater case. ♦
32. It will _ facilitate the reading of large numbers
not separated mto periods, if we begin with the units of
comparison, and proceed onwards to the left, saymg at
the fi,« digit ;' units," at the second " tens," at the
tlnrd 'hundreds," &c., marking in our mind the deno
mination of i\iQ highest digit, or that at which we stop.
We then commence with the highest, express its number
and denommation, and proceed in the same way with
each, until we come to the last to the right hand. ■
.+ !;^''^^^P;'J=— J^et it be required to read off 6402. Lookins
ti t;^'tlT"'^VV'^^r ^^y """^*^' ' '^' theO, ''tens;i
?,fv^ fi ' V^^"^,^*«.<i«i" .a^d at tlie C, thousands. The
.it.„er, therefore, being six thousands, the next diit is f(,nr
hundrod., &o Consequently, six thousands, fourlunu r ds
lV^ rf two units; or, briefly, six tliousand four hun
drcd and two, is the reading of tlie given number.
"m" ^^'t^ "'''^ ^^° ^•^^'^ *'^ facilitate notation, 'i^he
pupil will fust write domi a number of Doiit^ds of cyphers
26
NOTATION AND NUMERATION.
to roprosenf, tlio places to be occupied by the varioua
orders of imit.s. Jle will then put the digits express
ing the diflerent denominations of the given number
under, or instead of those cyphers which are in corres
ponding positions, with reference to the decimal points
bogmumg with the highest.
ExAMPLE.Write down three t'nousand rlx hundred and
bttytour The highest dent ■< • ^a being thousands, will
occupy the fourth place to thr ' ' the deoimal point. It
will be enough, therefore, to ^r down four cyphers, and
under them the corresponding digitsthat expressing the
thousands under the fourth cypher, the himdreis under tho
third, the tens under tho second, and the units under the
hrst; thus
0,000
3,654
A cypher is to be placed under any denomination iu
which there is no significant figure.
Example.— Set do^vn five hundred and seven thousand,
and sixtythree. '
000,000
507,063
After a little practice the periods of cyphers will
become unnecessary, and the number may be rapidly
put down at once.
34. The units of comparison are, as we have said,
always found in the first place to the left of the
decimal point ; the digits to thr left hand progressively
increase in a tenfold degree—those occupying the first
place to the left of the units of comparison being tea
times greater than the units of comparison ; those occu
pying the second place, ten times greater than those
which occupy the first, and one hundred times greater
than tho units of comparison themselves ; and so on.
Moving a digit one place to the left multiplies it by
ten, that is, makes it ten times greater ; moving it two
places multiplies it by one hundred, or makes it one
hundred times greater ; and sc of the res^ If all the
digits of a quantity be moved one, two, &c., places to
the left, the whole is increased ten, one hundred, &c.,
times— as tho case may be. On the other hand, moving
i
NOTATION AND NUMERATION.
87
' the various
'^its express
fen number,
re iu corres
uial point—
hundred and
ousands, will
lal point. It
3yphora, and
f pressing the
ids under tho
ts under the
mination ia
sn thousand,
yphers will
be rapidly
have said,
eft of the
'ogressively
ig the first
. being ten
those occu
than those
ics greater
md so on.
plies it by
i^ing it two
kes it one
If all the
, places to
idred, &c.,
ad, moving
a digit, or a quantity one place to the right, divides ifc.
by ten, that is, makes it ten tim^s smaller than before ;
moving it two places, divides it by one hundred, or
niiikes it one hundred times smaller, &c.
So. We possess this power of easily increasing, or
diminishing any number in a tenfold, &c. degree, whetlicr
the digits are all at the right, or all at the loft of the
docimal point ; or partly at the right, and partly at the
loft. Though we have not hitherto considered quautitiea
to tho left of the decimal point, their relative value will be
very easily understood from what wo have already said.
For the pupil is now aware that in the decimal system
the quantities increase in a tenfold dej^ree to the loft,
and decrease iu the same degree to "the right ; but
there is nothing to prevent this decrease to "the right
from proceeding beyond the units of comparison, and
tho decimal point ;— on the contrary, fioni the very
nature of notation, we ought to put quantities ten times
loss than units of comparison one plice to the right of
them, just as we put those which are ten times less than
hundreds, &c., one place to the right of hundreds, &c
We accordingly do this, and so continue the notation
not only upwards, but downwards, calling quantities U
the left of the decimal point integers, because none of
them is loss than a whole " unit of comparison :" an^
those to the right of it decimals. When there are deci
mals m a given number, the decimal point is actuallv
expressed, and is always found at the ri.<(ht hand side
ot the units of comparison.
30. The quantities equally distant from tho unit of
comparison bear a very close relation to each other
which IS indicated even by the similarity of their names •
those which are one place to the left of the units of com
parison are called " tens," being each identical with «r
equivalent to ten units of comparison; those which are
one place to the riffht of the units of comparison tim
called tenths," each being the tenth part of, tlint is, ten
times as small as a unit of comparison ; quantities two
places to the Ipfi of the units of comparison are called
imndrcds" being one hundred times greater ; and
those two places to the rigAl, " hundredths," beincr one
S8
h )••;
■M
NOTATION AND NUMERATION
hun red times loss t u.n tlio units of comparison ; and .a
ot .11 11.0 o hers to tl.c right and loft. This will bo mo o
evident on in.sp.,cting the following tabic :
Asrcirling Scries, or Integers
Uiic Liiit
Ilumlred . . jqq
Thousand . . 3,000
Ion tliou,s!ui(.l . 10,000
Hundred tlioiisand 100^000
&c.
II'
IVsccmlin.' Sories, or DccliiMls.
Ouo Unit.
Tenth.
Htindrcdtli.
Tiioijsiindth.
Tentliousfindth.
Hundred thousandth.
&G.
•1
•01
■001,
•000,1
•000,01
We have seen that when we divide integers into periods
Ifih.^ .separating point must be put to the rid.t
of the thousands; m dividing decimals; the first poin
nius^ be put to the right of the thousandths. ^
37. Oare must be taken not to confound what we
Tl "ul !^''^'^': ^v'itli what wo shall hereafter des I!
not irlent ica lly the same quantitiesthe decimals beino
wha sha 1 bo termed the " quotients" of the cone.^
pondmg decimal ft.actious. This remark is made hero ^
i^n''? %T'"''''"'^^" ''^'^ ^^^ ^^' subject, in those
who already know something of Arithmetic
T'x. '^^"^l^ ^'^ ^'^ ^■'^''^"^^ ^"^'' treating integers and deci
mals bj dfent rules, and at ditForc;^t tin!es, since the^
follow precisely the sa.ne laws, and constitute parts of
Uie very same series of numbera ]]esides, any quantity
nay, as far as the decimal point is concerned, be ex
pressod m diflerent ways; tbr this purpose ^e ha;e
Jierely to change the unit of comparison. Thus let it
be required to set down a number indicating five hun
dred and seventyfour men. If the " unit of compari
son ho one man, the quantity would stand as follows,
&74. If a band of ten men, it would become 574— f^r
as each man would then constitute only the tenth pa,J
of the "unit of comparison," four men would be only
ourtenths, or 04 ; and, since ten men would forn. bu^
one unit, seventy men would be merely seven uni^s of
comparison, or 7 ; &c. Again, if it wen. a band of one
ImM mm, the number must be ^vritten 574 ; an]
I'lbiij, ir a miiKi ol a Inuusand 9mij it would be 0574
m
NOTATION AND NUMERATION.
29
son ; and so
will be more
, or DooJiiwls.
:h.
Ith.
siindth.
•tliousfindth.
Into periods
to the rJLdit
first point
d what we
after dosici;
equal, but
inials beiniT
the corres
;ide here to
t, in those
and deci
sinco they
3 parts of
y (juantity
ed, be ex
' wo have
'hus, let it
'l five him
conipari
ts follows,
374— for,
enth pact
1 bo only
form but
I units (»f
lid of one
■74 ; and
be 0574
Should the " unit " bo a band of a dozen, or a score
men, the change would be still more complicated ; as,
not only the position of the dechual point, but the very
digits also, would be altered.
39. It is not necessary to remark, that moving the
decimal point so many places to the left, or the digits
an equal number of plaees to the right, amount to the
same thing.
Sometimes, in changing the decimal point, one or
more cyphers are to be added ; thus, when we move 42 '(3
three places to the left, it becomes 42600 ; when wo
move 27 five places to the right, it is '00027, &c.
40. It follows, from what we have said, that a deci
mal, though less than what constitutes the unit of com
parison, may itself consist of not only one, but several
individuals. Of course it will often be necessary to indi
cate the " unit of comparison,"— as 3 scores, 5 dozen, 6
men, 7 companies, 8 regiments, &c. ; but its nature does
not affect the abstract properties of numbers ; for it is
true to say that seven and five, when added together,
make twelve, whatever the unit of comparison may be :—
provided, however, that the sa7?ie standard be applied to
both ; thus 7 men and 5 men are 12 men ; but 7 men
and 5 horses are neither 12 men nor 12 horses.; 7 men
and 5 dozen men are neither 12 men nor 12 dozen men.
When, therefore, numbers are compared, &c., they must
have the same unit of comparison, or — without alterin^r
their value — they must be reduced to those which have"?
Thus we may consider 5 lens of men to become 50
individual men— the unit of comparison being altered
from ten men to om man, without the value of tho
quantity being changed. This principle must be kept
m mind from the very commencement, but its utility
will become more obvious hereafter.
EXAMPLES IN NUMERATION AND NOTATION.
JVoiaiion.
1. Put down one hundred and four
2. One tliousand two hundred and forty
3. Twenty thousand, three hundred and fortyfive
^iu.i.
104
1,240
20,345
^
i !
mV.
: t
i'
i
•so
NOTATION AND NUMERATION.
5.
G.
7.
Two Imndrod and tliirtyfuur thousand, fivo
hundred and sixtyseven
Three hundred and twentynirio tliousand,
seven hundred and sevcnlynine
Seven hun(hed and nine tliousand, eight hun
dred and twelve . . *! .
Twelve hundred and fo/tysoveu tliousand,
four hundred and tiCtysovon
8. One million, three hundred and ninetyseven
thousand, four hundred and seventylive
0. Put down fiftyfour, seventenths
10. Ninetyone, fivo hundredths .
11. Two, threetenths, four thousandths, and four
hundredthousandths
12. Nino thousandths, and three hund]ed thou
sandths • • . . .
13. Make 437 ten thousand times greater
14. Mako 2 7 one hundred times greater
15. Mako 0056 ten times greater . .
10. Make 430 ten times less
17. Mako 275 one thousand times les^i .
Jln.i.
234,507
320,771)
709,812
1,217,457
1,397,475
547 •
0105
2*30401
000903
4,370,000
270
056
43
000275
Numeration
7. read 8540320
5210007
Gi.)30405
50 0075
3' 000000
00040007
2. — 407 8.
3. — 2700 • 0.
4. — 5000 10
6. — 37054 n.
G. — 8700002 12.
13. Smnd travels at the rate of ahout 1142 feet in a
seeond ; light moves ahout 195,000 miles in the same time.
14. The sun is estimated to be 880,149 miles in diameter:
its size is 1 377,013 times greater than that of the earth.
15. Tho diameter of the planet mcreurv is 3,108 miles,
and his distance from the sun 30,^; 14,721 miles.
10. The diameter of Venus is 7,498 miles, and her dis
tance from the sun 08,791,752 miles.
_17. The diameter of the earth is a])out 7,904 miles: it is
95,000,000 miles from tlie sun. and travels round the latter
at the rate of upwards of 08,000 miles an hour.
18. The diameter of the moon is 2,144 miles, and her dis
tance from the earth 230,847 )niles.
10. The diameter of Mars is 4,218 miles, and his distance
from the sun 144,907,030 miles.
20. The diameter of Jupiter is 89,009 miles, and his dis
tance from tho sun 494,499,108 m.ilcs.
I
flvO Jtnt.
. 234,507
and,
. 320,771)
liiiii
. 709,813
ami,
. 1,217,457
3vcn
. 1,397,475
547 •
0105
four
. 2.30401
tiou
. 000903
. 4,370,000
270
050
43
. 000275
W32C)
10007
M)405
0075
oooon
040007
^42 feet in a
e same time,
i in diameter j
the earth.
! 3,108 miles,
I.
and her dis
1 miles : it in
md the latter
, and her dis^
I his distance
, and his dis
NOTATION AND NUAIKRATION.
31
)
21. The diameter of Saturn is 78,730 miles, and hia die
taiico from the sun 907,089,032 miles.
22. 'J 'ho length of a pendidmu which would vibrato
Hcconds at the ci^uator, is 39011,084 inches; in the latitude
of 4o degrees, it is 39116,820 inches; and in the latitude of
90 degrees, 39221,95G inches.
23. It has been calculated that the distance from the
earth to the nearest fixed star is 40,000 times the diameter
of the earth's orbit, or annual path in the heavens ; that is,
about 7,000,000,000,000 miles Now suppose a camion
ball to fly from the earth to this star, with a uniform velocity
equal to that with which it first leaves the mouth of the
gun~say 2,500 feet in a second— it would take nearly
1,000 years to reach its destination.
24 A p.iece of gold equal in bulk to an ounce of water,
would weigh 19258 ounces; a piece of iron of exactly the
sanie size, 7788 ounces; of copper, 8788 ounces; of lead.
11'352 ounces; and of silver, 10474 ounces.
NoTK.— The examples in notation may be made to answer
for numeration ; and the reverse.
QUESTIONS IN NOTATION AND NUMERATION.
[The references at the end of the questions show in what
paragraphs of the preceding section the respective answers
are principally to be found.]
1. What is notation } [1].
2. What is numeration .? [1].
3. How are we able to express an infinite iriety of
numbers by a few names and characters ? [31 .
4. How may we suppose ideas of numbers to have
been origmally acquired > [4, &c.].
^5. What is meant by the common ratio of a system
of numbers .> [12] . j ^^
6 Is any particular number better adapted than
another for the common ratio .? [12]. ■
ratio .?^[11]^^''' '^'*'''^' ""^ numbers without a common
8. What is meant by quinary, decimal, duodecimal,
vigesimal, and sexagesimal systems ? [13].
9. Explain the Arabic system of notation ? ri5"I
10. What are digits .? [161 ■^'
1 1 TT *l 1
. 1 . ^ow arc tliey maac to express all numberg ? [17] .
I
■^^ ti
S9
NOTATION AND NUMERATION.
■*or of units of a lower order precisely the same thing ?
14. Have the characters wo use, always and every
Inhere been cmp oyed to express numbers ? [211 ^
rh.!iff ♦ ''i^' ^r^"^^^ P^>""*' «^»d the posit on of
figiJesfe]."' ''^ '^ °^'"« affect significant
F^n'T,^''?^ !? the difference between the English and
aolLIns^l '^^ '*^"^° '^^^^^'^ *^g ^
«.i^; T^'''* \' "'"''''* H *^^ ascending and descending
X.^ [36] ' '" "^ "'" the/ related to each
n„fJ' f '''V*^^* i". expressing the same quantity, we
mist place the decimal point differently, according to
the unit of comparison we adopt ? [38]
22. What effect is produced on a digit, or a quantity
by removing it a number of placs to the right, or left
or similarly removing the decimal point ? [34 iid 39]
^f.
le and relativo
33
luivalent num
le samo thiujr ?
ays and every
' [211.
:on? [22, &c.].
lie position of
ICG to it ? [26
3ct significant
e English and
)eriod8? [29 J.
integers and
nd descending
slated to each
quantity, we
according to
or a quantity
right, or left,
[34 and 39]
SECTION 11.
THE SIMPLE RULES.
SlftlPLE AUDlTiON.
oc,'; tLv""t'"ir "•'""«"'' J'y ""7 arithmotioal pro
ocas, they are cither morcasod or (iimiDislii.r) • if \„
f?™toe££SS:?
but vo may have m mmm of tl,™ , ""^ """^'
called "Multiplieator''\?l Zv'' ™ ^l'""'/'' '"
.««, but Jheir'n„„.b?r'i. imlS^t' Ty™ I^^Jf Z
quantities to be UodL^lnlL'T' "" 1";'"'"='' "^ *«
til" kind l,„t ..„r.i ' '""'''Pl'ca'ion restricts us as to
.•eally comprehended under trtl' ™^^SS' ""
£.s^rn^i:t^r=:
^..o%fu^';^iftiret;f.i,;L7;r^^^
means, that G is to be S d'od "o 8 wf "^ ^ ^ "" ?'
prcfacd, the positive is undtstood '"" "" "^^ ^
to 16. ' ^^'^ ^'^^ ''^^" "^ 9 i^^^ 7 is equal
II, ^if
34
ADDITION.
Quantities connected by the sign of addition, or that of
equality, may be read in any order ; thus if 7 + 9=16, it
is true, also, that 9 + 7=16, and that 16=7+9, or 9+7.
5. Sometimes a single horizontal line, called a viii
mlum, from the Latin word signifying a bond or tie,
is placed over several numbers ; and shows that all the
quantities under it are to be considered, and treated as
\)\xt one ; thus in 4+7=11, 4 + 7 is supposed to form
but a single term. However, a vinculum is of little
consequence in addition, since putting it over, or remov
ing it from an additive quantity — that is, one which has
the sign of addition prefixed, or understood — does not
in any way alter its value. Sometimes a parenthesis ( ) is
used in place of the vinculum; thus 5+6 and (5+6)
mean the same thing.
6. The pupil should be made perfectly/ familiar with
these symbols, and others which we shall introduce as
we proceed ; or, so far from being, as they ought, a
great advantage, they will serve only to embarrass him.
There can be no doubt that the expression of quantities
by characters, and not by words written in full, tends
to brevity and clearness ; the same is equally true of the
processes which are to be performed — the more con
cisely they are indicated the better.
7. Arithmetical rules are, naturally, divided into two
parts ; the one relates to the setting down of the quan
tities, the other to the operations to be described. We
shall generally distinguish these by a line.
To add Numbers.
RirtE. — T. Set down the addends under each other,
so that digits of the same order may stand in the same
vertical celumn — units, for instance, under units, tens
under tons, &c.
II. Draw a line to separate the addends from their
mm.
III. Add the units of the same denomination together,
boci:!nninf» at the rijzlit linnd side.
IV. 9 the sum of any column bo less than ten, set it
down under that column ; but if it be greater, for every
ADDITION.
35
s from their
ten it contains, carry one to tlie next column, and Dut
down only what remains after deducting the'tens^tf
'wthing remains, put down a cypher '
V. Set down the sum of the last column in full.
8. Example.— Find the sum of 542375^984—
375 } addends.
984 J
1901 sum.
Tt j'c . ' '^' ^"*^ ^' which are "hundreds" in nnnthoiT
JO £t^pr:s^i,/SS
Vn5? ^'""Pf' ^'ly"'" ^^^™^^ notation, can easily find
l7d"Krof " ': ^" ' ^^^"" ^^^^r 5 sini all
tiic dio ts that express it, except one to the ri^vht hnnrl
j^ide, ml indicate the number of '4ens'' it JonVt „s
thus m 14 there are 1 ten, and 4 units • in li^ S J '
d 2 units ; in 143, 14 tens, and's unit's, L'' ' *"''
The ten obtained from the sum of the units alon<r wUh «
:;reds._„„d write down a cypher in theTns. pLe" rf S^^
The two hundreds to be "carried " idflorl +a o Q i k
As there are no thousands in the next cnlumn ^h.i •
ti.o last clnm i^'full. ""' ™*' ™ '<'' •^"™ "'« »"■» «f
«lSri^Un<SLSimr^;rlL!
!i;!i
36
ADDITION.
that we may easily find those quantities which are to be added
together ; and that the value of each digit may be more clear
from its being of the same denomination as those which are
under, and over it.
Reason of II.— We use the separating line to prevent the
sum from being mistaken for an addend.
Reason of III.— We obtain a correct result only by adding
units of the same denomination together [Sec. I. 40] :— hun
dreds, for instance, added to tens, would give neither hnndreda
nor tens as their sum.
We begin at the right hand side to avoid the necessity of
more than one addition; for, beginning at the left, the process
would be as follows —
542
375
984
1,700
190
11
1,000
800
100
1
1,901
The first column to the left produces, by addition, 17 hun
dred or 1 thousand and 7 hundred ; the next column 19 tens,
or 1 hundred and 9 tens ; and the next 11 units, or 1 ten and
1 unit. But these quantities are still to be added :— beginning
again, therefore, at the left hand side, we obtain 1000, 800 100°
and 1, as the respective sums. These being added, give 1,901
as the total sum. Beginning at the right hand rendered tho
successive additions unnecessary.
• ^\f^T'^^ OF IV.— Our object is to obtain the sum, expressed
m the highest orders, since these, only, enable us to represent
any quantity with the lowest numbers ; we therefore consider
ten of one denomination as a unit of the next, and add it to
those of the next which we already have
After taking the " tens » from the sums of the different
columns, we must set down the remainders, since they are
parts oiihQ entire sum; and they are to be put under the
CO umns tliat i.roduced them, since they have not ceased to
Dclong to the denominations in these columns
Reason or V.It follows, that the sum of the last column
ot„ \ ff * ^""y""^ '!" ^''^^ ' ^'^^' ('" *he above example, for in
it contains ""'' '" "°*^^"g *o be added to the tens (of hundreds)
10. Proof of^Addition.—Oxxi off the upper addend,
by a separating line ; and add tlie sum of tho (uantitie»
ADDITIOIf
87
re to be added
be more clear
ose which are
;o prevent the
mly by adding
I. 40] :~hun
ther hundreds
e necessity of
ft, the process
tion, 17 hun
iumn 19 tens,
, or 1 ten and
I : — beginning
000, 800, 100,
id, give 1,901
rendered the
im, expressed
to represent
fore consider
und add it to
the diflferent
Qce they are
lit under the
lot ceased to
3 last column
™ple, for in
9f liundreda)
lor addend,
(uantitie»
under, to what is above this line. If all the additions
have been correctly performed, the latter sum will be
equal to the result obtained by the rule : thus—
5,673
4,632
8,697
2,543
21,545 sum of all the addends.
15,872 sum of all the addends, but one.
5,673 upper addend.
21,545 same as sum to be proved.
This mode of proof depends on «ie fact that the whole in
equal to the sum of its parts, in whatever order they are
Sf i' • Ji'*i' ^'""^l^. ^° *^® objection, that any error com
nutted in the first addition, is not unlikely to be repeated in
the second, and the two errors would then conceal each other
To prove addition, therefore, it is better to go through
the process again, beginning at the top, and proceedinc
downwards. From the princ^le on whicS the JtS of
proof IS founded, the result of both additionsthe direct
and reversed— ought to be the same.
It should be remembered that these, and other proofs of
shfcf U ?, nnl '• ^^''"'^ '"'''^? "" ^'^^ d«g^«« «f pro'babiUty,
since It is not in any case quite certain, that two errors cal
culated to conceal each other, have not been committed.
ul'i^'^^^^^ Qwaw^2Vw!5 containing Decimals. —From.
What has been said on the subject of notation (Sec. I.
db) It appears that decimals, or quantities to the right
hand side of the decimal point, are merely the continu
ation, doivnwards, of a series of numbers, aU of which
to low the same laws ; and that the decimal point is
mtendpd not to show that there is a difference in the
nature of quantities at opposite sides of it, but to mark
ni f !i^v ""'^ 1 «TP""^°^" ^ Pl^^^d. Hence the
mJo tor addition already given, r.pplies at whatever side
a I, or any of the digits in the addends may be found
it IS neceaary to remember that the decimal point in
the sum should stand precisely under the decunal points
of the ^a.lends ; smce the digits of the sum must beffrom
the very nature of tl^ — p°"=, rm a uT,oui.,num
,, . ■' ''^'; '•' ^!' inuce.^3 [D , ot exactly the same
value, rospectirely, as the digits of the addends under
g2
38
ADDITION.
whicli thoy are ; antl if set down as tlioy should bo, their
denominations are ascertained, not only by their position
with reference to their o i decimal point, but also by
their position with reference to the digits of the addends
above them.
Example.
263785
460602
637 008
6263
1887695
It is not necessary to fill up the columns, by adding
cyphers to the last addend ; for it is sufficiently plain
t^at we are not to notice any of its digits, until we come
to the third column.
12. It follows from the nature of notation [Sec. I.
40], that however we may alter the decimal points of
the addends— provided they are all in the same vertical
column— the digits of the sum will continue unchanged ;
mus in the followin<y : —
4785
8257
6546
14588
4785
3257
6546
14588
4785
3257
6646
14588
•4785
•3257
•6546
1^4588
•004785
•003257
•006546
•014588
I:
EXERCISES.
(Add the following numbers.)
Addition,
Multiplic
(1) (2)
(3)
(4) (5)
4 8
3
6 4
6 4
9
6 4
3 7
7
6 4
6 6
6
6 4
7 2
5
6« 4
~~ —
—
— —
_ —
—
— —
(10)
(11)
(12)
6763
3707
2867
2341
2465
8246
5279
5678
1239
(6)
9
9 CO
9
9
9
(13)
6978
3767
1236
Involution,
(8)
4
Ttt i
(14)
5767
4579
1236
tO "
(9)
5
6
5
5
u
(15)
7647
1239
3789
m
ADOITJON,
uld be, their
ilieir position
but also by
the addends
(16)
6673
123?
2345
(17)
8767
4567
1^34
(19)
5147
3745
6789
(20)
34567
47891
41234
39
(21)
73456
4567?
9123J
3, by adding
iiently plain
itil we come
ion [Sec. I.
al points of
ame vertical
unchanged ;
•004785
•003257
•006546
•0X4588
(22)
(23)
(24)
(25)
(26)
(27)
76789
34567
78789
34676
73412
36707
46767
89123
01007
78767 
70760
46770
12476
45678
34667
45679
47076
36767
(28)
(29)
(30)
(31)
(32)
(88)
45697
76767
23456
46678
23745
87967
87676
45677
78912
91234
67891
32785
36767
76988
34567
66789
23456
64127
(34)
(35)
(36)
(37)
(38)
(39)
30071
45667
45676
37412
37645
67456
47656
12345
76767
12345
45676
34567
12345
37373
12315
67891
37676
12345
47676
45674
67891
10707
71267
67891
lution.
8)
^4
4
4
4
«5 "
(9)
5
5
5
5
 u
(15)
7647
1239
3789
(40)
71234
12498
91379
92456
(46)
87376
12677
88991
23478
(41)
19123
67345
67777
88899
(47)
78967
12345
73707
12(371
(42)
93456
13767
37124
12156
(48)
34567
12345
7776G
67345
(48)
45678
34567
12345
99999
(19)
47676
12345
67671
10070
(44)
45679
34567
12345
76767
(50)
67678
12345
67912
4G7()7
(45)
76766
34567
12345
67891
(51)
67667
34567
23456
76799
40
ADDITION.
(52)
76769
12346
76776
466G6
(53)
57667
19807
34076
13707
(64)
767346
4767 of
467007
123456
(55)
478894
767367
412346
671234
(56)
876767
123764
845678
912346
(67)
676
4689
87
84028
(58)
74564
7674
376
6
(69)
5676
1667
63
6767
(60)
76746
71207
100
. 66
(61)
67674
76670
36
77
(62)
4237
6684
2793
6241
(63)
087
6273
8127
?563
(64)
03786
20766
00253
10004
(66)
86772
603482
578563
71262
(66)
00007
06236
•0572
•21
(67)
64718
66347
2160:^
000007
(68)
810235
37603
47125
653712
(69)
00007
5000
427
3712
(70)
84536
37
8456302
•007
(71)
57634
4000005
2135
2763
m:
72. £7654 + £50121 + £100 + £76767 4 £67^5
=£135317.
73. £10 + £7676 + £97674 + £676 + £9017
=£115053.
74. ^971 +£400+£97476+£30+£7000+£76734
=£18261 1.
T5. 10000 + 76567 + 10 + 76734 + 6763 + 676741
=176842. r"/D/M
76. 1 + 2 + 7676 + 100 + 9 + 7767 + 67=15622
/7. 76 + 9970 f 33 + 9977+100 f 67647 + 676760
=764563. rJiy>iuu
ADDITION.
41
(67)
676
4589
87
84028
(68)
087
6 273
8127
?563
(67)
ri8
3347
21602
000007
(71)
57634
000006
2135
753 •
+ £675
 £9017
■£76734
6767+1
=15622.
676760
m
78. 75 + 6 + 756 + '7254 +'345 +'5 +'005 +07
• =37514.
79. •4+7447+37007+7505+747077=:934004.
80. 5G054475 + 007+3614+4672=101619.
81. •76 + 0076 + 76 + 5 + 5 + 05.=82'3176.
82. •5 + 05i005+5 + 50 + 500=:555'555.
83. •367+567+762 + 976+471==1387667.
84. 1+1 + 10 + '01 + 160+001=17M11.
85. 376 + 443+476lt55=52966.
86. 3677+4'42+M001 + 6=428901.
87. A merchant owes to A. £1500 ; to B. £408 ; to
0. £1310 ; to D. £50 ; and to E. £1900 ; what is the
sum of all his debts ? A7i,s. £5168.
88. A merchant has received the following sums : — .
£200, £315, £317, £10, £172, £513 and £9 ; what ia
the amount of all ? Ans. £1536.
89. A merchant bought 7 casks of merchandize. No.
1 weighed 310 tb ; No. 2, 420 ft ; No. 3, 338 ft ; No.
4, 335 ft ; No. 5, 400 ft ; No. 6, 412 ft ; and No. 7
429 ft : what is the weight of the entire ?
Ans. 2644 lb.
90. What IS the total weight of 9 casks of goods :
Nos. 1, 2, and 3, weighed each 350 ft ; Nos. 4 and 5,
each 331 ft ; No. 6, 310 ft ; Nos. 7, 8, and 9, each
342 ft .? Am. 3048 ft.
91. A merchant paid the following sums : — ^£5000,
£2040, £1320, £1100, and £9070; how much was
the amount of all the payments ? Ans. £18530.
92. A linen draper sold 10 pieces of cloth, the first
contained 34 yards ; the second, third, fourth, and fifth,
each 36 yards ; the sixth, seventh, and eighth, each 33
yards ; and the ninth and tenth each 35 yards ; how
many yards were there in all .? Ans. 347.
93. A cashier received six bags of money, the first
hold £1034 ; the second, £1025 ; the third, £2008 ; tho
fourth, £7013 ; the fifth, £5075 ; and the sixth, £89 ;
how much was the whole sum .? Aiis. £16244.
94. A vintner buys 6 pipes of brandy, containin<r as
follows :— 126, 118, 125, 121, 127, and 119 galbns ;
how many gallons in the whole ? A/as. 736 gals.
95. What is the total weight of 7 casks, No. 1, con
42
ADDITION.
tainiug, om ib ; No. 2, 725 lb j No. 3, 830 ib • No 4
VaBib; No. 6, 6«7 1b, No. B, 609 ib; and No.' 7,'
Js como to r "'«'^1"'"<'"° «o^t ^39, what will 20
S auTetlT' "''fy«?^™i five thousand Lbm;
t,™, ^^^ ™' 'w thousand seven hundred anl
tvm>ty.one ; hftys,. thousand seven hundred and seven^
inn \,M n ■!,■ ^''" 206729644.
fo„. .•■n ""™.'"''I'««s and seventyone thousand ^
four „,dhons and cghtysix thousand ; two mil ionTMd
tweive" „£L a ' "'LXX^eno^u^lir, '™j
ventytwo thou.,and, „i„e\„nd e'd a"d twen tlr™^
s:^,^t^^l3^£d5:s^eSS^
four hundred and ninetyone thou.,and. ^^J. 3 8700o'
102 Add together one hundred and sixtvsevon tl,m,
dred 'a d'ii' V 'f T^r™ 'housatdTZ tt
fhi:;^; t ntruV, i'^^ntn^fdirrs
■tno ,\,n ,1 ^ ,, ^^^^'. 3665000.
iu,j. Add three tcntIionsan<ltlis • fo.tv fn,,,. r
tenth, ; live hundredths ; six .hou.a.Uths, ti'ltlenltl "i;!
ADDI'IION.
43
«andths ; four thousand aud forty K)no ; twcutytwo, one
tenth ; one tenthousandth. ' Ans. 4107*6r)72.
104. Add one thousand ; one tenthousandth ; five hun
dredths ; fourteen hundred and forty ; two tenths, three
tenthousandths ; five, four tenths, four tliousandths.
Ans. 24456544.
105. The circulation of promissory notes for the four
weeks ending February 3, 1844, was as follows : — Bank
of England, about iE21, 228,000 ; private banks of Eng
land and Wales, £4,980,000 ; Joint Stock Banks of
lOngland and Wales, ii;3,446,000 ; all the banks of Scot
land, £2,791,000 ; Bank of Ireland, £3,581,000 ; all the
other banks of Ireland, £2,429,000 : what was the total
circulation ? Ans. £38,455,000.
106. Chronologers have stated that the creation of
the World occurred 4004 years before Christ ; the deluge,
2348 ; the call of Abraham, 1921 ; the departure of the
Israelites, from Egypt, 1491 ; the foundation of Solomon's
temple, 1012 ; the end of the captivity, 536. This being
the year 1844, how long is it since each of these events ?
Ans. From the creation, 5848 years ; from the deluge,
4192; from the call of Abraham, 3765; from the de
parture of the Israelites, 3335 ; from tlic foundation of the
temple, 2856 ; and from the end of the captivity, 2380
107. The deluge, according to this calculation, occur
red ] 656 years after the creation ; the call of Abraham
427 after the deluge ; the departure of the Israelites,
430 after the call of Abraham ; the foundation of the
temple, 479 after the departure of the Israelites ; and
the end of the captivity, 476 after the foundation of the
temple. How many years from the first to the last ?
Ans. 3468 years.
108. Adam lived 930 years ; Seth, 912 ; Enos, 905 ;
Cainan, 910 ; Mahalaleel, 895 ; Jared, 962 ; Enoch, 365 ;
Methuselah, 969 ; Lamech, 777 ; Noah, 950 ; Shem, 600 ;
Arphaxad, 438 ; Salah, 433 ; Hebor, 464 ; Peleg, 239 ;
Eeu, 239 ; Serug, 230 ; Nahor, 148 ; Terah, 205 ; Abra
ham, 175 ; Isaac, 180 ; Jacob, l47. What is the sum of
all their ages ? . Ans. 12073 years
13. The pupil should not be allowed to leave addition,
u
ADDITIOIf.
until ho can with groat rapidity, continually ad.l any of
without hositation or furtL mention of the' numbtV
J or instance he Bhould not bo allowed to proceeHhus :
8 nr« ia"'' ^f i ^^ ""^ ' ^^^ 21 ' ^' J "«r even 9 a, ci
b are 16 ; and a are 21 ; &o. Ho shoJld be able, uTu
raately, to add the following— '
6638
4768
9342
1Q786
in this manner :2, 8 ... 16 (the sum of the column •
of which 1 IS to be carried, and 6 to be set down) " s!
10... 13; 4,11 ... 17; 10,14... 19. •' '
QUESTIONS TO BE ANSWERED BY THE PUPIL.
reduce^d'?^[7]"''''^'"^'''"'^"" those of arithmetic be
.2. What is addition .? [3J.
tion ? ^3^* ""^^ *^' ''''°''' '^ *^' quantities used in addi
t' WkI ^^\*^' ^S"? °^ ^^'^^*^'^°' ^^^ equality ? [41
ndditir;Utit rs"^^^ ^ ^^^ ^'^'  ^^^ ^^^^ 
^* Su^* ^^ *^*® ^'^^^ ^°^ addition ? [7]
7. What are the reasons for its different parts > [91
8. J^es this rule apply, at whatever side of the deci
fo^dT[llj"' "' '"^ '' *^^ ^"^"^^^^^« '' ^^ dded a"e
9. How is addition proved ? [10]. ^
10. What is the reason of this proof.? [10].
8UUTRACTI0N.
46
SIMPLE SUBTllACTION.
14. Simpk siibtraction is confined to abslract numbers,
and apphuate which consist of but one denomination
^subtraction enablcH us to take one number called' tho
subtrahend, from another called the minuend. If anv
tlnng s loft, it is called the excels ; in commercial con
cerns, It IS termed the remainder ; and in the mathema
tical sciences, the difertmce.
15. Subtraction is indicated by —, called the minus,
or negative sign Thus 54=1, read five minus four
equal to one, mdicates that if 4 is substracted from 6.
unity is left. »
Quantities connected by the negative sign cannot be
taken, indifferently, m any order ; because, for example,
64 is not the same as 4—5. In the former case the
positive quantity is the greater, and 1 (which means
+ L4J) IS left; m the latter, the negative quantity
18 tlie greater and 1, or one to be subtiactld, still
remams.^ To illustrate yet further the use and nature
ot the signs, let us suppose that we hmx five pounds
and owe four;— the five pounds we hate will be repre
sented by5 and our debt by 4 ; taking the 4 f?om
the o, we shall have 1 pound ( + 1) remaining. Next
let us suppose that we have only four pounds and owe
five ; If we take the 5 from the 4that is, if we pay
a. fkr as we cana debt of one pound, represented by
1, ^111 still remain ;— consequently 5—4=1 ; but
nr.^^' V'"'"''^"'? placed over a subtractive quantity,
or one having the negative sign prefixed, aiteTs its
value, unless we change_all the^igSs but the first •
thus 53+2, and 5—3+2, are not the same thing
^If ^t^^'* ^ but 53+2 (3+2 being considered
nowas but one quantity) =0 ; for 3+2=5 ;therefor«
j3+2 IS the same as 55, whifA leaves nothing ; or.
m herwords, it is equal to 0. If, however, we cli'an.e
all the ai'T'"° ^^^r^^* ai.~ x?_i .i i _ . ' ^""^,0
i uc sj^,.,.j vAvcpt ixic mKi, ine vaiuo of the quantity is
l^M
4R
••UHTRACTIOPI.
uot aUcio(l by flit) viiKMilmn ;— thus 53^2=4; and
f*~3 — 2, also, Is c(iual to 4.
Again, 27447— 3=27.
27
But
■4+73=19.
27—4—743 (chaiifyinip all tho ilgng of the ) OT
' ori((iuuI quantitloi, but tha flnt) { ■*■*• •
The following examplo will show how the vinculum
attecta numbers, according us wo mako it include an
additive or a subtractiv<3 quantity :
48f 738f72 =49.
4817 — 3 — 8 JL7_ 2=49 • what ia under the vinculum beinjf
' additive, it is not necessary to
, change any signs.
48f7— i5l8 — 712=49 • ^' " ""^ necessary to change aU the
d^_L7 q sr~"Tio ,n ',«.'«"» «»'^<'"tlie vinculum.
^'^"r'~'J— "— 7f2 =49; it is necessary in this case, alto,
48173817=2=49; u ^^^^J^^j^^,, ease.
In the above, we have sometimes put an additive, and
sometimes a subtractive quantity, under the vinculum ;
in the former case, wo wore obliged to change the signs
ot all the terms connected by the vinculum, except the
hrst— that IS, to change all the signs under the vin
culum ; m the latter, to preserve the original value of
the quantity, it was not necessary to change any sign.
To Subtract Numhers.
„nrl!" ^""'^TI: ^^^'' ;^'' ^^«^*« «f *^« subtrahend
under those of the same denomination in the minuend—
units under units, tens under tens, &c.
II. Put a line under the subtrahend, to separate it
iroin the remainder.
III. Subtract each digit of the subtrahend from tho
one over it m the minuend, beginning at the right hand
IV. If any order of the minuend be smaller than tho
quantity to be subtracted from it, increase it by ten : and
cither consider the next order of the minuend as lessoned
by unity, or the next order of the subtrahend as in
creased by it.
V. After subtracting any denomination of the sub
suiirriACTiOiV.
47
h2=:4; and
lie flnt) ! =27.
he vinculum
include an
vinculum hein^
ot aeoessary to
1.
to change all the
inciilum.
thia case, also,
ns.
in this case.
(Iditivc, and
I vinculum ;
;e the signs
except the
Icr the vin
al value of
any sign.
subtrahend
minuend —
separate it
1 from the
right hand
5r than the
J ten ; and
IS lessoned
end as in
f the sub
trahend from the correspoudiug pjut of the nunuond
H',;fc (l.)wii wh.it i,s loft, if liny thing, in the phiee which'
b<.'li)ii(;s to thi! Kaiiie donroiiiri.itioti (»f the " rcnuiindfr."
Vr. ]Jat if th(!rc \a no\\nn<^ Idl, put down a cyphoi
provided any digit of the " rcMuuiudor" will be niore dis
tant from the deciiuul point, and ut tlio same side of it.
18.. K.\A.MrM.: l.~8ubtr;ict 427 from 71)2.
n)2 minuend.
427 «ul;tnilu!iul.
u(J5 remainder, ditterenco, or ox<'(!,s«<.
Wo cannot take 7 units iunxx 2 unitfl; hut "bomnvin." nfl
It IS calind, one ot the !) tons in the )ninti(Mi(l, ,uid consi,],.,.
«ng It as /6'u unit.s. wo add it to the 2 units, and tbnn liavo
1 units; taking / Iroiii 12 unit.s, 5 arc left: wo put o in
the units pace o/ the "n^maindor." Wo may considor tlio
.' tens ot tho niinuend (one luiving been taken away, or
borrowed) as 8 tons; or, which is the same thin. 'may
suppose the I tons to remain as they wore, b.t tho''2 tcn"^
ot tho subtrahend to hi.ve beoomo [\; tlicn, 2 tens from «
t.ns or o tens from U ten.s, and tens are lell :wo i.ut i\
in the tens' place of tiie "remainder. 4 hundreds, of tho
Hubtrnbend, taken from the 7 Junidreds of the ininurud,'
ExAMPLK 2.— Take 5G4 from 7G8.
7G8
504
2U4
When G tons are taken from G ten.s, notliing is Irft : w,.
therefore put a cypher in the tens' place of the ^emainder"
KxAMi=j.E 3.— Tako 537 from 5U4.
594
537
s:
When 5 hundreds are taken from 5 hun.livds notliln..
48
SUBTRACTION.
Of the subtrahend maybe near those of the mmuend from
which they are to be taken ; wo are tlien sure that the JrL^
Sfriuff^Evfl*'' subtrahend and nainueuS ^^ybe
douMn. fo1>,n^^^ "^ arrangement, also, we remove any
1„ ^ \ , *^^ denominations to which the diVits of the sub
trahend belongtheir value? being rendered more cert^Lbv
S;SZ Tf rf''''' '' the'cligits of th'rnS"' '^
Keason OF H.The separating line, though convenient i
not of sueh importance as in addition [9] ; si^io th^" remain
der » can hardly be mistaken for another quantity
Keaso^t of Ill.When the numbera are considerable
powe"' ?tt"mii^^^^^^^^ be effected at once, from'ThflSd
poweis ot tne mind; we therefore divide the ffiven ouantitipq
into parts; and it is clear that the sum of the d'ffe?en?es of
th^ sTmTrS"/ ''T' ^f, ^"l^i^ *^ *» diffe're^r Ween
to 500 4nl7n 9^TS =7*^"'"' ^^^^27 is evidently equal
rebble^"^?"^ w7?.+^~J\r "•''? ^^ «h«^» *« the child by
be necesstrv tonU. ^'"^ ^* "ilP^^*. ^^^'^ ''^^' ^^cause it maj
oe necessaiy to alter some of the d g ts of the minuend so a*^
to make it possible to subtract from them the corresnondin^
ones of the subtrahend; but, unless we beSn at the STt h3
Bide, we cannot know what alteratiois may be iVquired ''''^
thin'tlfo'n '" '^T^' ",^^. ^"g^* '' tlie iinuend be smaller
than the corresponding digit of the subtrahend, we can proceed
l?on fTl f *•'" ""T. .*;"''*' ^« ^'^y i^^rease .hat denmina
tion ot the minuend which is too small, by borrowing <,nrfrom
or'tE v^^Ms t^obf'''^' ""'. 'r f t^e lower InlSaL^
p^^:ii; i^ tr^^. ^i^^^^r^it^ s
idrcds.
tens.
units.
7
4
8
2
12 = 792, the minuend.
/ = 427, the subtrahend
^ 5 = 305, the diflference.
an^'r,^wV' "I® "['^y «dd equal quantities to both minuend
woulfw '"'^' ^'^^"^ ^^" ^°^ ^^^' the difference; tien we
HuiKireds. tens.
7 9
4 2 + 1
6
units.
2 4 10 == 792 f 10, the minuend f 10.
7 = 42 7 f 10, tlie subtrahend f 10.
== 365  0, the same difference.
ini«!fKf"'''^ ""a T'!''*'?"? ^« ^0 not use the given minuend
and subtrahend, but others which produce the ^ImTvZZl^T,
Reason of V.— The remainders obtained bv subtraetin^
successivfilv. fl.o riiffn«„„4. .i„„_. •, ,. "'^  "/ suDtracting,
froai those which correspond in the minuend are the i^ari of
,1 ij,"
SUBTRAC'IION.
> minuend from
that the corres
linueud may be
WQ remove any
gits of the &ub
nore certain, by
s minuend.
1 convenient, is
ie the " I'cmain
tity.
3 considerable,
pom the limited
;iven quantities
e differences of
jrence between
Jvidontly equal
;o the child by
because it may
minuend, so as
s corresponding
the right hand
equired.
nd be smaller
we can proceed
ihat denoraina
iwing one from
denomination,
to those of the
we alter the
, in the exam*
uend.
trahcnd.
rence.
both minuend
Jnce ; then we
id f 10.
lend f 10.
ifforence.
iven minuend
no remainder,
subtracting,
ts Bubtrahend
i the jparts of
4Q
the total remainder. They are to be set down under the .Ipnn
20 Proof* of SvMradion.—Add to^.her the re
mainder and subtrahend ; and the mm sliould be equal
to the minuend. For, the remainder expresses by Lw
£hf' *t 'f ' r""^"^'^«r to the subtrahend
should make it equal to the minuend ; thus
8754 minuend.
6839 subtral. nd. ^
2915 difference. )
Sum of difference and subtrahend, 8754=.minuend.
wh?t^^tf^'^ */f /^"^^^^*»1*^^ f'*^"^ ^^^e minuend, and
f hp I ' 7 "^^1 ^' "'^"'^^ ^" ^^'' subtrahend. For
8G84 minuend. Pun^r. . aroA ^ i
rciQK 1 i 1 , irRooF : obrf4 minuend
i98o subtrahend. J549 remafnder.
649 remainder. New remainder, 7985=subtrahend
8034 minuend.
7985 subtrahend.
T,.„, , , 049 remainder.
Difference between remainder and minuend, 7986= subtrahend
T^mV' Tho^lT''*\'^'^'f'' *■". ^^'^^'^■f^^^^^ contain Deci^
s do "i^fl r • ^T^ ^''^'V ^PPli^'^t'J^, at whatever
k fold tv'TnP'"'^ '"^^ '' '"^"^'^^ ^^^« ^i^^^ts may
VMT '7.;' follows, as in addition [11], from the
Y'y n.Uuie of notation. It is necessary to put th.=
decimal point of the remainder under the IZS'JnU
v^lZ.lT! fl "; '""S ^^ ^^'""^ ^*"^^t, have the samo
^alue as the digits from which they have been derived.
60
Example.
SUBTRACTION.
Subtract 42785 from 50304.
66304
42785
13519
Since the digit to the right of the decimal point in the
onthT' ?■' ''"'^''''''' '"^"* '' ^'^'' ''^''' *^»« subtraction o i.o
t e inhtl n' T^""^ P?mt indicates vvimt remuins^fter
the subtraction of tlie units, it expresses so many units:
all this IS shown by the position of the decimal point. '
An^^'J\ ^y^'''^'' ^'''™ *^^ principles of notation [Sec. I.
40J, that however we may alter the decimal points of
the mmuend and subtrahend, as long as the/stand in
the same vertical column, the didts of the difference
are not changed ; thus, in the following examples, the
fiame digits are found in all the remainders •—
4362
8547
815
436
354'
816
4362
3547
815
•4362
•3547
•0815
•000 J 3 62
•0003547
•0000815
EXERCISES IN SUBTRACTION.
. From
Take
(1)
1969
1408
(2)
7432
6711
(3)
9076
4567
(4)
8146
4377
(5)
3176
2907
(fi)
76877
45761
Froi.i
Take
(7)
86167
61376
(8)
67777
46699
(9)
71234
43412
(10)
900076
899934
(11)
376704
297610
(12)
745674
376789
From
Take
(13)
67001
35690
(14)
9733376
4124767
(15)
567i)74
476476
(16)
473(i76
S21799
(17)
6310756
3767016
(18)
376576
240940
m
SUBTRACTION.
il point In the
)tiacti(m of tho
ice the digit to
i remsiiiis after
many units; —
al point.
:ation [Sec. 1.
nal points of
they stand io
;he difference
3xamples, tlie
3 : —
•0001362
•000;J547
•0000815
(19.
From 345070
Take ]799
(20)
234100
9U0
(21)
4367676
25G560
(22)
845073
I2479'J
(23)
70101076
37091734
61
(24)
67300000
31237777
From
Take
(25) (26) (27) (28) r29) no^
S?? JS f= ??S = ~
47134777 1123640 7476909
(31)
From 7045076
Take 3077097
(36)
From 11000000
Take 9919919
(82)
87670070
26716645
(37)
3000001
2199077
(33)
70000000
9999999
(38)
8Q00800
(J77776
(34) (35)
70040500 60070007
56767767 41234016
(39)
8000000
62358
(40)
404006b
220202
(5)
(6)
nQ
76377
i907
45761
I)
704
610
)
(12)
745674
376789
(18)
'50 370570
)16 240940
From
Take
(41)
8573
4216
(46)
From 000003
Take 000048
(42)
805 4
732
(47)
87432
663705
(43)
694763
85600
(48)
67004
23
(44)
47030
0078
(49)
47632
0845003
(45)
52137
20005
(50)
400327
00006
745676—507456=178220
500789—75074=501115.*
9410005007=935993.
9/001—50077=40024
70734977=75757.
56400100=50300.
700000—99=099901.
5700—500=5200.
9777—89=9088.
700001=75099.
900173=90014.
02. 977774=97773.
03. 000001=59999.
64. 75477—76=76401.
65. 797 105=692.
00. 175— .074=1676.
07. 9707—4709=92301.
08. /• 05— 4776=2274.
09. 10701—9001=176.
70. 12100097121=407909
n. 1701 — •007^..176093.'
72. 1500
< 803=7197.
53
SUBTRACTION.
73. What number, .iddcd to 9709, will make it 10901
oaJ: ^J^^*"*^^' ^"^1^t 20 pipes of hrandj, containing
2459 gallons, and sold 14 pipes, containing 1680 gal
Ions ; how many pipes and gallons liad he remaining ?
Ans. 6 pipes and 779 gallons.
75. A merchant bought 664 hides, weighin^r 16800
Jb, and sold of them 260 hides, weighing 78091b ; how
many hides had he unsold, and what was their wei^rht >
76.
Am. 304 hides, weighing 8991 lb
A gentleman who had 1756 acres of land, gives
2o0 acres to his eldest, and 230 to his second son ; how
many acres did he retain in his possession > Ans. 1276
77. A merchant owes to A. i^SOO ; to B. £90 • to D
7o0; toD. ^600. To moot tl.co' debts t has but
d;.971 ; how much is he deficient ? Ans jei269
78 Paris is about 225 English miles distant from
London; Eonie, 950; Madrid, 860; Vienna, 820
Copenhagen, 610; Geneva, 460 ; Moscoav, 1660 : Gib
ral^nr, 1160; and Constantinople, 1600. How much
more distant IS Constantinople than Paris; Rome than
Madrid ; and Vienna than Copenhagen. And how much
less distmit IS Geneva than Moscow; and Paris than
Madrid } Am. Constantinople is 1375 miles moro dis
taut than Pans; Rome, 90 more than Madrid; and
V lenna, 210 more than Copenhagen. Geneva is 1200
miles less distant than Moscow; and Paris, 635 less
tnan iuadnd.
79. How much was the Jewish greater than the
J^.ngIish mile ; allowing the former to have been 13817
miles Endish >
80.
mile ;
mile 5
81.
Am. 03817.
How much IS the English gieater than the Roman
allowing the latter to have been 0915719 of a
T#u'^' .1 , . ^^^ 0084281,
W hai IS the value of 6  3 + 1 5  4 .? Am. 1 4
Afis S'?
Of 4762+12441634 } Am. 52 94
84. What is the differencc betwccn 15+13—6—81 +
Am. 38.
S2. Of 43 + 73^^?
83.
84. __ ^
02, and 15+13—6=11 + 62 }
23.
Before leaving this rule, the pupU should "be able
;j
i
M
"*
MULTIPilCATlON.
53
:c it 10901
Alls. 1192,
5 containing
; 1680 gal
:nainiug ?
'79 gallons,
iing 16800
09 It) ; how
eir weight ?
ig 8991 lb.
land, gives
I son ; how
Ans. 1276.
£90 ; to C.
be has but
ns. jei269.
stant from
ma, 820 ;
660 ; Gib
low much
lome than
how much
Paris than
moro dis
Jrid ; and
a is 1200
, 635 less
than the
an 13817
. 03817.
iie Roman
)719 of a
)084281
Ans. 14
Ans. 33
IS. 52 94
•6—81 +
Ans, 38.
I be able
to take any of the nine digits continually from a given
number, without stopping or hesitating. Thus, sub
tracting 7 from 94, he should say, 94, 87, 80, &c. ; and
should proceed, for instance, with the following exampla
5376
4298
1078
m
this
manner :~8, 16.. .8 (the difference, to be set
down); 10, 17...7; 3,3...0; 4, 5...1.
QUESTIONS TO BE ANSWERED EV THE PUPIL.
1. What is subtraction .? [14].
2. What are the names of 1*e terms used in subtrac
tion ? [14].
3. What IS the sign of subtraction ? [15].
4. IIow is the vinculum used, with a subtractive
quantity? [16].
5. What is the rule for subtraction .? [17].
6. What are the reasons of its different parts.? [19].
7. Docs it apply, when there are decimals ? [211
tion
8. How is subtraction proved, and wliy .? [2oT.
9. Exemplify a brief mode of performiu'^ su
" ' [23] ^ °
subtrac
SIMPLE MULTirLIGATION.
24. Simple multiplication is confined to abstract
numbers, and apphcate which coataiu but one denomi
nation.
,».^?r ^^'?^'"" '"f '^'' "' f" ''^^^ ^ ^"^'^ity. ^'^^"cd the
^ ¥^anrl, a number of times indicated by the ,nM
ittl U ^''f'^'^''' .'% t'^^t h wlucli we multiply :
the x^sult m he multiplication is called the jJdt.
addo d,'' ,n multipheatiou, is termed the <' multipli '
JZh^ !^' '^^ '' designatoa tho " product.'' The
Uuautu.es which, whou muitipljcd log.^lcr, give tho
' ' J.)
m
Ml
64
MULTIPLICATION.
poduot, nro cnlle.l also factors, nnd, when they ,re
•S'^rtf^^'"* J^^^'^^ ;"/^^ ^ — than' two
actors m that case, the multiplicand, niultinlier or
b and 7, be the factors, either 6 times 6 may be con
^^V^^;/""Itiphcand, and 6 times 7 as the mulLLr
2o. Quantities not formed by the continued addS^on
of any number, but unitythat is, which are not h^
product^; of any two numbers, unle'ss unityTs taken as
one of themare called privii numbers : Yll ot ts are
termed co^nposiie. Thus 3 and 5 are p me but 9
and 14 are composite numbers; because oniyXJ
midtipbed },y...,,will p«pduce "'three," aU oKi
mu tip led by one, will produce " five, "but //S
multi^hed by three will produce " nine," ind seTeSi mil
tiphed by hvo will produce " fourteen "
fl!.r. \ ■*PT^'^ ^^ ^" e«^%eror, in other words
matder i^ f'^^r^f ^^"^^ ^^ ^^^^^^^ leaving a rei
contabed in I .1 "" '\ ^""'^^'^ "^ ^^^ ^'^^^'^ it i«
Pd from if a^t^b^^f r^^pieS b'yNt"
measure ot 14, because, taking t as often as r^n^^ihC.
from 14 4 will .till bo leftithis, I.3_3=,0, /oS
5=0, but 14—5=9, and 9—5=4.
Measure,
', ,:. , ' "," ^^ — ''^ry, una y — 5=4.
submultiple, and aliquot part, are synonymous.
is V numllrTrr V'''' '^ *^" ^^ ^^°r« q^^^tities
IS a number that will measure each of them • it is a
measure comnon to them. Numbers which^ have no
ocner all otliers are comj^^^z^e to each other. Thus 7
and 5 are i;r^/;^e to each other, for unity alone will
b::s:3':iiV ' ^^^^ ^^^ ^^^^^sr^ :^,^^
uecause 3 wUl measure either. It is evident that two
ITi ^Z^'l ""f "= p™ "> ^''' *" , ttfs
e°ceM „n^t. fr"' •""''''"'■'', •'"™"' "»• '^ ""•'■. ''"<'
MUI,TIPLICATIO^^
65
on they are
I'e than two
lultiplier, or
Thus, if 5
May be con
iiltiplier — oi
nultiplier.
led addition
ire not the
is taken as
I others are
ime, but 9
only tkreey
id onlyfivey
but, three
I seven mul
tne number
thor words,
ving a ro
ot pari of
3cause it is
Q be sub
d by 3, an
) is not a
IS possible
I, JO— 5r=
Measure,
•
quantities
1 : it is a
have no
'Me to each
Thus 7
tlone will
ch other,
that two
r thus 3
:o, and —
will niea
■"fti
Two numbers may bo oninpnsito to each other, and
yet (yne of them may be a fvme number ; thus 5 and 25
are both measured by 5, still the former is ytrim.
Two numbers may be composite, and yet prime to
emh other ; thus 9 and 14 are both composite numbers,
yet they have no covwion measure but unity.
28. The greatest common measure of two or more
numbers, is the greatest number which is their common
measure ; thus 30 and 60 are measured by 5, 10 15
and 30 ; therefore each of these is their ccmimon mea
sure ; — but 30 is their greatest common measure. When
a product is formed by factors which are integers, it is
measured by each of them.
29.^ One number i the i^uUiple of another, if it
contam the latter a number of times expressed by an
integer. Thus 27 is a multiple of 9, because it con
tams It a number of times expressed by 3, an integer
Any quantity is the multiple of its measure, and the
measure of its multiple.
^ 30. The com7non multiple of two or more quantities,
IS a number that is the multiple of each, by an intcffer •
thus 40 IS the common multiple of 8 and 5 ; since it is a
multiple of 8 by 5, an integer, and of 5 by 8, an integer.
^ LhY<^st common multiple of two or more quantities,
IS the /m5^ number which is their common multiple,
thus 30 IS a common multiple of 3 and 5 ; but 15 is
then least copunon multiple ; for no number smaller
tnan lo contains each of them exactly.
31. The equivmltiples of two or more numbers, are
then products, when multiplied by the same number ;—
tlius 27, 12 and IS, are equimultiples of 9, 4, and 6 •
because, multiplying 9 by three, gives 27, multiplying 4
^^4f 'tV^u'r^'.''^^ ^^^itiplying 6 by three, give^ IS.
S2 Multiplication greatly abbreviates the process of
addition ;— for example, to add 68965 to itself 7000 times
})y audition," would be a work of great labour, and con
sume much time ; but by " multiplication," as we shall find
presently It cn^i be done with case, in less than a minute.
roi.i 7 i?v "•''^ .'"'"' inaccurate, to have stated
L^J that multiplication is a species of addition ; since we
can know the product of t^vo quantities without havin^'
06
MULTIPLICATION.
recourse to that rule, if tliey are found in the multipli
cation table ?..t it must not be forgotten that the mul
plioation table IS actually the result of additions, long
since made ; without its assistance, to multiply so simplf
a number as 4 by so smaU a one as five, we shodd be
obliged to proceed as follows,
4
4
4
4
4
20
performing the addition, as with any other addends
The multiplicat on tabl^is due to Pythagoras, a" cele
bSortcS '''''^''''^ ^^^ ^ ^^ ^^^ ys
34. We express multiplication by X ; thus 5x7—
thatXT *^\^r^*fd by 7 ire equal tol's, ^^
that the product of 5 aTid 7, or of 5 by 7, is equal to 35
When a quantity under the vinculum 'is toTe muf^
pliedfor, to multiply the whole, we must multiply
eack of Its parts^:— tlms^^7+8=3=3X7+3xSl
3X3; and 4+5X8+36, means that each of the
terns under thelaiier vinculum, is to be multiplied by
each of those under the former ^ ^
maf be^ro^ltf ""'''''"' f ^^ ?^' ^^^n of multiplication
may be lead in any order; thus 6X6=6X5 This
Will be evident from the foUowing mustration, by which
of it :! ' ^ ' ^'^^^^^"S to the view we take
«>
8
♦ ♦ •
♦ ♦ •
♦ • •
» • »
"^ ^ » •
Quantities connected by the mgn of multipUcation,
MULTIPLICATION.
Jie multipli
bat the mul
tlitiona, long
ly so simple
e should be
67
Idends.
>ras, a cele
590 years
us 5x7=
I to 35, or
qual to 35.
) be multi
be multi
t multiply
r+3xs—
ch of the
Jtiplied by
ttiplication
<5. This
. by which
Bonsidered
y we take
V
't
plicati^
ion.
pre multiplied if we multiply one of the factors ; thus
GX7X3 multiplied by 4=6X7 multiplied by 3X4.
36. To prepare him for multiplication, the pupil
should be made, on seeing any two digits, to name their
product, without mentionhig the digits tiiemselves. Thus,
a largo number having been set down, he may begin
mth the product of the first and second digits; and
then proceed with that of the second and thhd, &c!
Taking
587C349258G7
for an example, he should say: — 40 (the product of 5
and 8) ; 56 (the product of 8 and 7) ; 42 ; 18 ; &c.,as
rapidly as he could read 5, 8, 7, &c.
To Mibltijply Nmibers.
37. When neither multiplicand, nor multiplier ex
ceeds 12 —
Rule. — Find the product of the given numbers by
the multiplication table, page 1.
The pupil should be perfectly familiar with this table.
^ Example.— What is the product of 5 and 7 ? The mul
tiplication table shows that 5x7=35, (5 times 7 are 35).
38. This rule is applicable, whatever may be the
relative values of the multiplicand and multiplier ; that
is [Sec. I. 18 and 40], whatever may be the kind of
units expressed— provided their ahsolwte values do not
exceed 12. Thus, for instance, 1200X90, would come
under it, as well as 12X9 ; also •0009X08, as well as
9X8. We shall reserve what is to be said of the man
agement of cyphers, and decimals for the next rule ; it
will be equally true, however, in all cases of multiplica
tion.
39. When tlie multiplicand does, but the multiplier
does not exceed 12 —
Rule. — I. Place the multiplier under that denomi
nation of the multiplicand to which it belongs.
II. Put a line under the multiplier, to seplirate it from
the product.
ni. Multiply each denomination of the multiplicand
by tiie multiplier— bogiuniii;;^ ut the rit^'ht hand side.
+1
08
MULTIPLICATIOX.
I\. If tlm prodiipt of tho multiplier and any digit
of tho liiultiphcand is Ichs than ten, set it down under
that (tigit ; but if it bo greater, for every ten it contains
carry one to tho next produ.., and ])ut down only what
remains, after d. o' u tir. the tens; if nothing remains,
put down a cypher. '
,V. Set down the last product in full.
40. Example. 1.— What ia the product of 897351x4?
SOTI^Sl multiplicand.
4 multiplier.
3581)404 product.
4 times one unit are 4 units; since 4 is less than ten, it
gives nothing to be "carried," we, therefore, Bet it down n
the units' place r f the product. 4 times 5 are twenty (tens):
which are equal to 2 tens of tens, or hundreds to I o carried,
and no units of tons to be set down in the tens' place of
the product— in which, therefore, we put a cypher 4
times 3 are 12 (hundreds), which, with the 2 hundids to bo
carried from the tens, make 14 hundreds; these are equal
to one thousand to bo carried, and 4 to be set down in the
thousan<l8' pluee of the product. 4 times 7 are 28 (thou
sands), and 1 thousand to be carried, are 29 thousands ; or
2 to be carried to tho next product, and 9 to be sot do\vn
4 times 9 are 3b, and 2 are 38 ; or 3 to be carrriod, and 8 to
be set down 4 times 8 are 32, and 3 to be carried are 35 ;
which 13 to be set down, since there is nothing in the next
denomination of the multiplicand.
Example 2.— Multiply 80073 by 2.
80073
2
16014G
Twice 3 units are units ; G being less than ten, gives
nothing to be carried, hence we put it down in the units'
place of the quotient. Twice 7 tens are 14 tens; or 1 ' undrod
to be carried, and 4 tens to be set down. As there are no
hundreds in the iiadtiplicand, we can have none in the pro
duct, except that whicli is derivtsl from the multiplication
ot the tens ; we accordingly put the 1, to be carried, in the
hundreds' place of the product. Since there are no thou
sands in the multiplicaud. nor any to be carried, we put a
cypher in that denomination of tho product, to keep any
significant iigures that follow, in their proper places.
i
i
MULTIPLICATION.
5U
41. Reason of I.— Tho multiplier ia to ))o pluccd under that
dcnominfiMon of the multiplicaud to wliicli it belouRs; sinco
tliere is tJien no doubt of its vhIuo. Sometimes it is necessary
(0 add cypiiers in putting down the muitiil:er ; thu.s.
EXAMPI.E 1.— 478 multiplied by 2 liundred—
47H multiplicand.
200 multiplier.
Example 2.539 multiplied by 3 ten thousandths—
68'J • multiplicand.
00003 multiplier.
Reason of II.— It ia similar to that given for tlio separatinff
line in subtraction [10]. ^ e
Reason ov III.— Wlien tho multiplicand exceeds a certain
amount, the powers of the mind are too limited to allow us
to multiply it at once ; we therefore multiply its parts, in suc
CQSsiun, un.l add the results as wo proceed. It is clear that
tho sum of the products of the parts by the muliinlior, is equal
to the product of tho sum of tlie parts by the same multi
plier :— tlius, 537x8 is evidently equal to 500 x8f;;0x 847x8
For multiplying all the parts, is multiplying the wliole ; since
the whole is equal to the sum of all its parts.
We begin at the rigl.^ hand side to avoid the necessity of
athnimrds adding together the subordinate products Thus
taking the example given above ; were wo to begin at the left
liand, the process would be —
897351
4
3200000=800000x4
360000= 90000X4
28000^ 7000x4
1200= 300x4
200= 50X4
4= 1x4
3589404=8um of products.
iV^Z^ri °^ jy:~^'r'^,*^'® ''""^ ^'' ^^''"^^ "^f "'« fourth part of
the rule for addition [9]; the product of the multipl/er and
any denomination of the multiplicand, being equivalent to the
bum of a colur. n m addition. It is easy to change the oiveu
ixainp e to an .xercise in addition; for 807851 x I, is theime
thing as
897351
897331
897351
897351
3589404
m
J
fl
60
hlULTirUCATlOS.
RKABopr OF y.It follows, that tho Inflt pro^liict h to be eot
<lown in ful; tor tlie tens it contains will not bo incroaseU :
they in«y, tlioroloro, bo sot down at once.
This riilo includca all casos in wlilnli tho ahsolii/e
value ^ of the di^'its in the luultiplior d.xs not excoea
, 12. Their rcdativo value is not niatoriii ; for it is as
easy to multiply by 2 thousands as by 2 units.
42. To prove multiplication, wluni tho mnltiplier dooa
not exceed 12. Multiply the multiplionnd by th(> mul
tiplior, minus one ; and add the multlplicjind to the pro
duct. Tho sum should bo the same as the product of
tho multiplicand and multiplier.
Example.— Multiply G432 by 7, and prove tho iviult.
C432 multiplicand.
6=7 (the multiplier) ~1
6432 3S502 multiplicand xO.
7(=C+1) 0432 multiplicand Xl.
45024 = ' 45024multipllcandmultip]iodby 0,1=7.
We have multiplied by 0, and by 1, and adtlod the results ;
but SIX times the multiplicand, plus once the multiplicand,
IS equal to seven times tho multiplicand. What we obtain
from the two processes snould be the same, for we Wve
merely used two methods of doing one thino.
EXERCISES FOR THE PUPIL.
Multiply
Bj
(1)
76762
2
(5)
763452
6
(9)
866342
11
(2)
67450
2
(6)
456769
7
(3)
78976
6
(7)
854709
8
(4)
57340
6
Multiply
By
(8N
45678f
?
Multiply
By
(10)
788679
12
(H)
476387^
11
(12)
fa>t29763
12
MULTIPLICATION.
61
5t ?fl to be Bot
)o incronsed :
;ho ahsolnfe
not (jxcood
for it is as
Itiplier dooa
ly the HI al
io tlio pro
product of
I'Viult.
th» reanlts ;
ultiplican<l,
i we obtain
3r we Wve
(4)
57040
6
43. To Multiply when the Quantities contain Cyphers,
or Dmrnals. — Slie rules alroady given aro applicable ;
those which follow aro consetjuonces of them.
When thoro arc cyplicrs at the cud of tho multipli
cand (cyphers in tho middle of it, Lavo been already
noticed [40])—
Rule. — Multiply as if there were none, and add to tho
product as many cyphers as have boon neglected. For
Tho greater tho quantity multiplied, tho grontor ought to
be tho product.
Example. Multiply 5G000 by 4.
5C00O
4
224000
4 timoa imita in tho fourth place from the decimal point,
arc evidently 24 ixnits in the same place ; — that is, 2 in tha
fiflh place, to be carried, and 4 in the fourth^ to be set down.
That wo may leave no doubt of the 4 being in tho fourth
}>lace of" tho ;or()duct, we put three cyphers to tho rij^ht
land. 4 times G are 20, and tho 2 to be carried, make 22.
44. If tho multiplier contains cyphers —
Rule. — Multiply as if there were none^ and add to
the product as many cyphers as have been neglected.
Tho greater the multiplier, tho greater the number of times
the multiplicand is added to itself; and, therefore, the greater
the product.
ExAMPLK.— Multiply 507 by 200.
5G7
200
113400
From what we have said [35], it follows that 200x7 is
the same as 7x200 ; but 7 times 2 hundred are 14 hundred ;
and, consequently, 200 times 7 are 14 hundred ;~that is, 1
in tho fourth place, to be carried, and 4 in the third, to be set
down. We add two cyphers, to show that the 4 is in the
third place.
45. If both multiplicand and multiplier contain
cyphers —
Rule. — Multiply as if there were none in either, and
add to the product as many cyphers as are found in
both.
d2
m
62
MULTirLICATiON.
Each of the quantities to be multiplied adcla cyphers to tho
product [43 and 44].
Example. Miihiply 46000 hy 800.
40000
800
50800000
_ 8 times G thousand yrocld bo 48
times six thousand ought to prod
thousand
number
8 hundred
100 times
greater— or 48 hundred thousand ;— that is, 4 in the scvcnt/i
place from the decimal point, to be carried, and 8 in tlie
are required.
But, 5 cyph
xLxtk place, to be set down.
to keep the 8 in the sixth place. After ascertaining the
position of the first digit in the p.^duct— from what the
pupil already knows— there cjin be no difficulty Avith tho
other digits.
46. When there are dechnal places in the multipli
cand —
Rule. — ^i\Iultiply as if there were none, and remove the
product (by nieails of the decininl point) so many places
to the right as there have been docuuals neglected.
Tlie smaller the quantity multiplied, the loss the product
Example.— Multiply 507 by 4.
567
4
2208
4 times 7 hundredths are 28 hundreths :— or 2 tcntlis, to
bo carried, and 8 hundredth « — or 8 in the second place, to
the right of tho decimal point, to be set down. 4 times 6
tenths are 24 tenths, which, with the 2 tenths to be carried,
make 20 tenths ; — or 2 units to be carried, and G tenths to
bo set down. To show that tlie re[>rosents tenths, we put
the decimal point to tho left of it. 4 times 5 units are 20
wiits, wliicli, with the 2 to ])e carried, make 22 units.
47. When there are decimal:? in the multiplier —
Rule. — Multiply as if there wore none, and remove
the product so many places to the right as there are
decimals in the multiplier.
The smaller tho quantity by which we multiply, the less
must be the rwult.
)hers to tlio
: 8 hundred
100 times
the seucnt/i
d 8 in tlie
e required,
iiining the
1 what tlie
7 Avith tho
3 muUipli
•cmove the
any places
cted.
product
I tcntlis, to
d place, to
4 times 6
he carried,
3 tenths to
the, we put
nJtH are 20
nits.
ier —
id remove
there are
y, the less
Example.
MULTIPLICATION,
Muiaply 5Go by 07
503
007
63
3941
3 multiplied hj 7 hundredths, is the same [351 as 7 hun
dredths multiplied by 3 ; whioh is equal to 21 hundredths : —
or 2 tenths to be carried, and 1 hundredth — or 1 in the
second place to the right of the decimal point, to be set down.
Of course the 4, derived from the next product, must be 07ie
place from the decimal point, «;c,
48. When there are decimals in both multiplicand
and multiplier —
Rule. — Multiply as if there were none, and move
the product so many places to the right as there are
decimals in both.
In this case the product is diminished, by the emallnesB of
both multiplicand and multiplier.
Example 1.— Multiply 563 by 08. *
563
•08
4504
8 times 3 tenths are 2*4 [46] ; consequently a quantity
one hundred times less than o — or 08, multiplied by three
tenths, vrill give a quantity one hvmdred times less than 24—
or 024 ; that is, 4 in the third place from the decimal point,
to be set dowTi, and 2 in the second place, to be carried.
Example 2.— Multiply 563 by 0 00005.
563
000005
00002815
49. When there are decimals in the multiplicand, and
cyphers in the multiplier 5 or the contrary —
Rule. — Multiply as if there were neither cyphers
nor decimals ; then, if the decimals exceed the cyphers,
move the product so many places to the right as will be
equal to the excess ; but if the cyphers exceed the deci
mals, move it so many places to the kft as will be
equal to the excess. '
The cyphers move the product to the left, the decimals to
the right ; the effect of both together, therefore, will be equal
to the difference of their separate effects.
64
MULTIPLICATION.
PI
ExAxMPLE 1.— Multiply 4600 bv "06
4000 ^
006 2 cyphers and 2 decimals J excess =0
276
Example 2.— Multiply 4763 by 300.
4763 "^
300
2 decimals and 2 cyphers; excess =0.
14289
Example 3.— Multiply 852 by 7000.
_J^^ 1 decimal and 3 cyphers ; exce8fl=2 oji>Men
596400
Example 4.— Multiply 57836 by 20.
57835
^^ _ 2 decimals and 1 cypher; excess =1 decimal.
115672
Multiply
By
EXERCISES FOR THE PUPIL
(13) (14)
48960 76460
5 9
(15)
678000
8
(16)
57d00
6
Multiply
By
(17)
7463
80
(18)
770967
900
(19)
147005
4000
(20)
661*76748
SOOOO
Multiply
(21)
743560
800
(22)
534900
SOOOO
(23)
60000
300
(24)
86000
6000
Slultiply
By
(25)
62736
o
(26)
8 7563
4
(27)
•21875
(28)
00007
8
MULTIPLICATION.
05
ss —
Multiply
By
(29)
5G341
00003
(30)
85G37
0005
(31)
721*58
00007
(32)
217G38
006
3=0.
2oyj!i.ewi
L decimal.
(16)
67000
6
(30)
o6{t76748
30000
(24)
86000
5000
(28)
00007
8
Multiply
By
(83)
875432
004
(34)
78000
03
(35)
51721
GOOO
(36)
3*^
000007
•00224
In the last example we are obliged to add cyphers to the
product, to make up the required number of decimal places.
50. When both multiplicand and multiplier exceed
12—
KuLE.— I. riace the digits of the multiplier under
those denominations of the multiplicand to which they
belong.
II. Put a line under the multiplier, to separate 'u from
the j)roduct.
III. Multiply the multiplicand, and eack part of the
multiplier (by the preceding rule [39]), beginning With
the digit at the right hand, and taking care to move the
product of the multiplicand and each sncce.ssive digit
of the multiplier, so mnny places more to the left, than
the preceding pioduct, as the digit of the multiplier
winch produces it is more to the loft tlian the signifi
cant figure by which we have kusi multiplied.
IV. Add together all tlie products; and their sum
will be tlie product of the multiplicand and multiplier.
51. ExAMPLi:.— Multiply 5634 by 8073.
5034
8)73
lG002=prodact by ?,.
39438 =pro(lact l)y 70.
45072 =product by 8000.
45483282=product Ijy 8073.
The product of the nuiUiplicand by 3, requires no e^i^
nation. ^
66
MULTIPLICATION.
7 tens times 4, or [35] 4 times 7 tens arc 28 tens : — 2 hun
dreds, to be carried, and 8 tens (8 in the second place from
the decimal point) to be set down, &c. 8000 times 4, or 4
times 8000, are 32 thousand : — or 3 tens of thousands to be
carried, and 2 thousands (2 in the fourth place) to be set
down, &c. It is unnecessary to add cyphers, to show the
values of the first digits of the different products ; as they
are sufficiently indicated by the digits above. The products
by 3, by 70, and by 8000, are added together in the ordicy
way.
52. Reasons of I. and II. — They are the same as those
given for corresponding parts of tlie preceding rule [41].
IIEASON OF III. — We are obliged to multiply successwely
by the parts of the multiplier ; since wo cannot multiply by
the whole at once.
xlEAsoisr OF IV. — The sum of the products of the multipli
cand by the parts of the multiplier, is evidently equal to the
■product of the multiplicand by the wliole multiplier ; for, in
the example just given, 5634 X 8073 = 5684 X 8000 f 70 f 8=
[34] 5034 X 8000+5634x7015634x3. Besides [35], we may
consider the multiplicand as multiplier, and the multiplier as
nmltiplicand ; then, observing the rule would be the same
thing as multiplying the new multiplier into the diiFerent
parts of the new multiplicand 5 which, we have already seen
[41], is the same as multiplying tlie whole multiplicand by
the multiplier. The example, just given, would become
8073X5634.
8073 new multiplicand
5684 new multiplier.
We are to multiply 3, the first digit of the multiplicand, by
6634, the multiplier; then to multiply 7 (tens), the second
digit of the multiplicand, by the multiplier ; &c. When the
multiplier was small, we could add the different productti as
we proceeded; but we now require a separate addition, — whicii,
however, does not affci the nature, nor the reasons of the
process.
53. To p'ove multiplication, when the multipliei ex
ceeds 12 —
EuLE. — Multiply the multiplier by the multiplicand ;
and the product ought to be the same as that of the
multiplicand by the multiplier [35] . It is evident, that
we could not avail ourselves of this mode of proof, in tho
last rule (b[42j ; as it would have supposed the pupil to
be then able to multiply by a quantity greater than 12
th
de
lei
in
of
Til
ini
7
ha
or
'■/
aol
«
th(
'4
Tal
1
bc^
1
'I
3 : — 2 hun
3laco from
lies 4, or 4
iauda to be
) to be set
show the
i] as they
e products
e ordicy
e as those
[41].
uccessively
aultiply by
e multipli
jual to the
er ; for, in
f 70+8=
)], we may
ultiplier as
the same
e diiFerent
ready seen
plicand by
Id become
plicand, by
the second
When the
)roductt as
n, — whici.,
ions of the
tipliei ex
tiplicand ;
lat of the
dent, that
oof, in tho
B pupil to
• than 12
MULTIPLICATIOINr.
67
mR
54. We may prove multiplication by what is called
" casting out the nines."
Rule.— Cast the nines from the sum of the digits of
the multiplicand and multiplier ; multiply tie remain
ders, and cast the nines from the product :— what is now
left should be the same as what is obtained, by cast
ing the nines, out of the sum of the digits of the product
of the multiplicand by the multiplier.
I^.XAMPLE 1. — Let the quantities multiplied be 942G and
'I'aking the nines from 9426, we get 3 us remainder.
And from 3785, we get 5.
47130
75408 3x5=15, from which 9
C5982 beino; taken,
28278 • 6 are left.
Tiiking the nines from 35077410, 6 are left.
The remainders l)e!ng equal, we are to presume tlie
multiplication is correct. Tlxe same result, however, would
liave been obtained, even if we had misplaced digits, added
or omitted cyphers, or fallei. into errors which had counter
acted each other : — with ordinary care, however, none of
these is likely to occur.
ExAMPr.K 2.— Let the numbers be 70542 and 8436.
T:>„king the :aincs from 76542, the remainder is G.
Taking them from
8436, it is 3.
459252
229626" 6x3=18, the
306108 remainder from which is 0.
612336
Taking the nines from 645708312 also, the remainder is 0.
Tho remainders being the same, the multiplication may
be considered right.
Example 3.— Lot the numbers be 403 and 54. ^
From 463, the remainder is 4.
From 54, ',■ ' 
1852 4x''=0 from which the remainder is 0,
23 15
From 2bd02 the remainder is 0.
u
kl
%
68
MULTIPLICATION.
Tlie remainder being in each case 0, wo arc to suppose
that the multiplication is correctly performed.
This proof applies whatever be the position of the
decimal point in either of the given numbers.
55. To understand this rule, it must be known that
a number, from which 9 is taken as often as possible,
will leave the same remainder as will be obtained if 9
be taken as often as possible from the Bum of its di.^its "
Since the pupil is not supposed, as yet, to have learned
divinon, he cannot use that rule for the purpose of
casting out the nines ; nevertheless, he can easily
ellect this object. •^
K !f, o'" f^7^'\ ^^""^^'er be 5C3. The sum of its digits is
+,.+'^' ^hile the nvimber itself is 500fG0+3.
First, to take 9 as often as possible from the sum of //,«
(hgrls. 5 and 6 aro'll ; from which, 9 being taken, 2 are
loit. ^ and 6 are 5, which, not containing 9, is to be set
down as the nmainder.
Next, to ta^o 9 as often as possible from the mmbcr itself..
503^=500 + 00+3=5 xl00+Gxl0+3=5x9iq^+Gx
9+1+3,= (if we remove the vinculum [34]), 5x99+5+
Ox.i+b+3 But any nnmber of niacs, will be found to he
f tie product of the same number of ones by 9 .—thus 999—
111X9; 99=11x9; and 9=lx9._Hence 5x99 express;^
a certain number of ninesbeing 5x11x9 ; it may, there
lure, be cast out; and for a similar reason, Gx9: after wliich
there will then be left 5+G+3from w'hic^i the luues are
still to be rejected; but, as this is the sum of the dibits we
must, in casting the nines out of it, obtain the same remain
aer as before. Consequently "we get the same remainder
whether we cast the mnes out of the number itself, or out
of the sum of its digits."
Neither the above, nor the following reasoning can
offer any difficulty to the pupil who has made himself
as fainiiar with the use of the signs as he ought :
they will both, on the contrary, serve to show how much
simpbcity, is derived from the u«e of characters express
ing, not only quantities, but processes ; for, by nieanj
ot such characters, a long series of argumentation mav
be seen, as it were, at a single glance.
5G "Costing the nines from the factors, n.iripiyina tU
resulting remainders, and casting tlie nines from thiH product,
MULTIPLICATION.
09
to suppose
ion of the
nown that
8 possible,
aincd if 9
its digits."
ve learned
urpose of
3aa easily
;s digits is
sum of it.<i
ken, 2 are
to he set
mhcr itself:
XOOfSf
und to be
hus 999=
expresses
I ay, tlioro
tov Mliich,
nines are
digits, we
fe remain
■emainder
elf, or out
ning can
himself
)ught : —
3W much
eypress
y means
ion ma;i
yinj> tli<»
product,
will Iftave the same remainder, as if the ninps were east from
the product of the factors," — provided the multiplication
has boon rightly performed.
To bhuw tliis, set down the quantities, and take away the
nines, as before. Let the factors be 573x464.
Casting the nines from 5J743 (which we have just seen
is the same as casting the nines from 573), wo obtain 6 as
remainder. Casting the nines from 4fGj4, we get 5 as
remaiiuler. Multiplying 6 and 5 we o btain 30 as product ;
which, being equal to 3x10=3x941=3x043, will, when
the nines are taken away, give 3 as remainder.
Wii can show that 3 will be the remainder, also, if we
cast the nines from the product of the factors ; — which ia
clFected by sotting down this product ; and taking, in suc
ecssion, quantities that are equal to it — as follows,
573x404 (the product of the factor8)=
SxBO+T xlO+S X 4 x 10046 xl0}4=
5x99flf7x9+l43 X 4 x 994146 x94l44= »
5x994547x947+3 X 4 x 994446 x9464.
5x09, as we have seen [55], expresses a number of nines;
it will continue to do so, when multiplied by all the quan
tiiies under the second vinculum, and is, therefore, to be
cast out; and, for the same reason, 7x9. 4x99 expresses
a number of nines ; it will continue to do so when multiplied
by the quantities under the first vinculum, and is, therefore,
to be cast out; and, for the same reason, 6x9. There will
then be left, only 54743 X41644, — from which the nines
are still to be (Tast out, the remainders to be multiplied together,
and the nines to bo cast from their product ; — but we have
done all this already, and obtained 3, as the remainder.
EXEnCISES FOR THE PVPII..
Multiply
By
(37)
765
^ 765
(38)
732
456
(39)
997
845
(40)
767
347
I'jtroducts *
Multiply
By
(41)
657
789
(42)
456
791
(43)
767
789
(44)
745
741
rroducts
1
■
70
MULTIPLICATrON.
.«n J' u! r "" ^yP^ors, or decimals in the multinli,
cand, rnultrpher, or botli ; the same rules apply as when
the niultipliar does not exceed 12 [43, &c.]
(1)
4600
67
(2)
2784
620
(3) (4)
3268 7856
26 032
(5) (6)
8796 482000
220 037
2G2200 1726080 84968 261892 193512 178340~"
Contractions in Multiplication.
r.,f;ioT^^° '!i'' °«* necessary to have as many deci
aTd\&"lier' ^"'"'*' ^^ "^ ^" ^^^^^ ^^^^P^^^
un?erM;r^r'''? *i' multiplier, putting its xmlis^ place
oZ? I' 1^ ""^i ^^, *^^^* denomination in the multipli
cand, which IS the lowest of the required product. ^
^.^'iW ^^.^^«h digit of the multiplier, beginninff
With the denomination over it in the multipl cand ; Tuf
addmg wha would have been obtained, on multip yW
the precedmg digit of the multiplicandunity, if^the
nmnber obtained would be between 5 and 15  2 if
between 15 and 25 ; 3, if between 25 and 35 ; &c
frt^Kl }7^^^ ^T^'''''^'''''^ ^^ *^^« products, aripinff
from the diflFerent digits of the multiplicand, stand ia
the same vertical column. '
Add up all the products for the total product; from
which cut off the reqmred number of decimal places.
59. Example 1.— Multiply 5G784 bv 97324 sn oa +«
have four decimals in the priuct ^^^^^24, so as to
Short Method. Ordinary M.thod.
56784
42379
511056
39749
1703
113
22
55^2643
567r t
97324
22i7136
1131568
1703i.")2
39748 8
51105G
'4
55 2044 601 (]
i j^LM.
MULTIPLICATION.
71
9 in the multiplier, expresses units ; it is therefore put
ander tho/o«r</i decimal pliioo ol'tho multiplicand— that being
tho place of the lowest decimal required m the product.
In multiplying by each succeeding digit of tho multiplier,
we neglect an additional digit of the multiplicand; because,
as tlie multiplier decreases, the number multiplied must in
crease—to keep the lowest denomination of the ditterent pro
ducts, tho same as the lowest denomination required in tlie
total product. In the example given, 7 (the second digit of
the multiplier) multiplied by 8 (the second digit of the mul
tiplicand), will evidently produce the same denomination as 9
(one denomination higlier than tbo 7), nniltiplied by 4 (one
denomination lower than the 8). Were we to multiply tho
lowest denomination of the multiplicand by 7, we should get
[4(5] a result in iha Jift/i place to the right of the decanal point ;
which is a denomination supposed to bo, in the present in
stance, too inconsiderable for notice— since we are to havo
only four decimals in the product. But we add unity for
evt'.ry ten that would arise, from the multipl cation of an ad(".
tional digit of the multiplicand ; since every such ten consti
tutes one, in tlie lowest denomination of the required product.
When the multiplication of an additional digit of the inulti
pliciind would give more than 5, ..ud less than 15 ; it is nearer
to the truth, to suppose we have 10, than either 0, or 20 ; and
thereiore it is more correct ta^add 1, than either 0, or 2 When
It would give more than 15, and less than 25, it is nearer to
the truth to suppose we havo 20, than either 10, or SO ; and,
therefore it is more correct to add 2, than 1, or 3; &c Wa
may consider 5 either as 0, or 10 ; 15 eil/ier as 10, or 20 ; &c.
On inspecting the re.sults obtained by the abridged,
find ordinary methods, the difference is perceived to bo
inconsiderable. When greater accuracy is desired, wo
should proceed, as if we intended to havo more decimals
in tho product, and afterwards reject those which are
unnecessary.
EvAMPLE 2.— Multiply 87653^ by 5704, so as to hav»
6 decimal places.
Mt 876532
^ 4G75
4383
G13
62
3
6051
,s^i
lil'ri
:_. ai 31! :U
72
MULTIPLICATION.
Tliere arc no units in tho multipljpr; but, as the rule
dirocts, wo put its units' place under tli^ third decimal place
ol the nmltipHcjuii. In multipivlng by 4, since there is no
di;,'it over it iu Lho multiplicand, we merely set down what
would have resulted from multiplying tho precodiu«r dono
mmatlon of the multiplicand. ° ^ a
Example 3.M.dtiply 4737 by 6731 so as to have
docnnal places in tiio product.
•47370
137G
284220
33159
1421
47
•318847
_ Na have put ^.he units' place of the multiplier under tho
suth decimal place of tho multiplicand, adding a cypher, or
su^iposing it to be added.
Example 4.— Multiply 84G732 by 0050, sc as to have
lour decimal places. ,
84 0732
G5
4234
508
•4742
Example S.—Multiply 23257 by 243, so as to have four
decimal places.
23257
342
465
93
•05G5
AVe are obliged to place a cypher in the product, to mako
up the required number of decimals,
00. To multiply by a Composite Number —
KuLis.— Multiply, successively, by its factors.
I
8 the rule
inial ^lacu
tlioro i no
ovnx what
iiug dcno
to have 6
[inder the
sypher, or
} to have
liave four
to malcG
i
MULTIPLl ATION.
73
EyAMPT.K— Multiply 732 by 90. 90=8x12' Lhotoforo
732x1"'
732 x8x 12.
732
8
[35
I
5850, product by 8.
12
70272, pi .act by 8x12, or 90.
If we multiply by 8 only, Ave multiply by a quantity 12
tinios too Hinall ; ami, therefore, tlie product will bo 12 times
loHH than it slioul ' We rectify tliis, by making the product
J J times greater— ...at ia, we multiply it by 12.
fU. When the multiplior is not exactly a Composltcf
Nui!i))or —
lluLi:. — Multiply by the factors of the nearest com
posite ; and add to, or subtract from the last product,
>s{) many times tlie multiplic \ as the assumed compo
site is le.ss or greater than the given multi2)licr
Example 1. — Multiply 927 by ^7.
87 = 7 X 1243 ; therefore 927 X 87 = 927 X 7x 12+15 =
927x7x12 + 927x3. [34].
927
7
0489:
12
:927 X 7.
77808 = 027x7x12.
2781 = 927x3.
80049 = 927 X 7 x 12 + 927 X 3, or 927 x 87.
If we multiply only by 84 (7 X 12), we take the number to
bo multiplied o times less than we ought ; this is rectified, by
adding 3 times the multiplicand.
ExAMPLK 2.— Multiply 432 b y 79. 79 = 812=9 x 92;
thoroforo 432 X 79=432 x 9 X 92=432 x 9 x9432x 2!
432
9
3888 = 432x9.
9
34992=432x9x9.
804=432x2.
34128=432 x 9 x 9432 x 2, or 432 x 79.
IMAGE EVALUATION
TEST TARGET (MT3)
1.0
I.I
.25
am
2.2
1^ illlU
Ik ..ill
IIIIIM
u mil 1.6
^.
%
<9
/^
*
'c^l
^^ >
^ cf
^^#^5^'
.**
:>/'
^^
HiotograDhic
Scieices
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 8724503
iV
^^^
•1>^
:\
\
..'" '^
'^> ^
/^
v. "^n
■^^f^^
l?.r
74
MULTIPLICA7I0K.
fli nifn K '^ r^° ^^ ^}'. *''° composite number, we have taken
Jh7product ^ subtracting twice the multiplicand from
62. This method is particularly convenient, when the
muKipher consists of nines.
To Multiply by any Number of Nines,—
liuLE.— Kemove the decimal point of the multipli,
cand so many places to the right (by adding cyphers if
necessary) as there are nines in the multiplier • and
subtract the multiplicand from the result. '
.ExAMPLK.— Multiply 7347 by 999 '
^ 7347 X 909 = 73470007347=7339053.
We, in such a case, merely muliinlv hv the ««»* i.* ^,
convenient composite number,'and "Sra^I the muHiputnJ
:xr;je^Ju«tS;4r ^^^ ^^ * ^^^en; thuftrh'a
7347x999=7347xi000l=:73470007347=7339653.
63 We may sometin.cs abridge multiplication hv
oonsidermg a part or parts of the multiplier as pro
duced Dy multiplication of one or more other parts.
ExAMPLK Multiply 57839208 by 62421648. The mnl.
tipUer may be divided as follows :0, 24, 216 and 48
24 = 6x4
216 = 24x9
48 = 24x2
57830268, iimltiplicand
62421648 , multiplier.
''S??oP3^ • = P''0(^"«t1>y 0(60000000).
I.ih8142432 : : product by 24 (2400000^
12493281888 : product by 216721600/
2776284864 product by 48. ^"
3610422427073664 product by 62421048.
,1 ^i'lP^o^?"^^ by 6 Avhen multiplied by 4 will dve the nrn
trn\ Y 24; the product by 24, Multiplied ^9, iill eive^th;
product by 216and, multiplied by 2. tlie prodactTy 4!!
fJt' Ji'f ''''.'\^^ !^^ difficulty in finding the places of
the first digits the difierent products. For when thoro
are neither cyphers nor decimals in the multiplicand—
ndthorTiS n;ultiplication, we may suppose that there are
neithei [4b, &c.]— the lowest douomination of each pro
MULTII'LICATION.
75
duct, will be the same as the lowest denomination of ths
multiplier that produced it ; — thus 12 units multiplied
by 4 units will give 48 units ; 14 units m.ultiplied by 4
tens will give 56 tens ; 124 units midtiplied by 35 units
will be 4340 units, &c. ; and, therefore, the beginning of
each product — if a significant figure — must stand under
the lowest digit of the multiplier from which it arises.
When the process is finished, cyphers or decimals, if
necessary, may be added, according to the rules already
given. ^
The vertical dotted lines show that the places of the lowest
digits of the respective multipliers, or those parts into which
the whole multiplier has been divided, and the lowest digits
of their resulting products are — as they ought to be — of the
same denomination.
48 being of the denomination units, when multiplied into
8 units, will produce units; the first digit, therefoi'e, of the
product by 45 is in the units' place. 216, being of the deno
mination himdreds when multiplied into units will give hun
dreds ; hence the first digit of the product by 216 will be in
the hundreds' place, &c. The parts into which the multi
plier is divided are, in reality,
COOOOOOO
2400000
• 21600
48
.=62421648, the whole multiplier.
We shall give other contractions in multiplication
hereafter, at the proper time.
EXERCISES.
45. 745X456^:=339720.
46. 476X767=365092.
47. 345X579=199765
48. 476X479=228004.
49. 897X979=878163.
60. 4 •59X706=3235 95.
61. 767X407=312169.
52. 467 X 606= '276942.
53. 700X810=567000.
54. 670X910=009700.
55. 910X870=791700.
66. 50014x70=350098.
57. 64 001X40=2560 04.
68. 91009X79=7189711.
59. 40170X80=3213600.
60. 707X604=427028.
61. 777 X •407=318239.
62. 7407X4404=32620428.
63. 6767X1307=7537469.
64. 67 •74X 1706=11 556444
65. 4567X2002=9143134.
66. 7767x3012=23394204
67. 9600X7100=68160000.
68. 7800X9100=70980000.
69. 6700X6700=44890000.
70. 5000X7600^38000000.
71. 70814x90l07=6380837098.
72. 97001X70706=7440658706.
73. 98400X07407=6295813800.
74. 56007x45070=2524235490
70
WULTIPLrCATION.
aolmtir"^''""'"^' '" ^1395; a pound being
7b. In 2480 pence how many farthings : four far
thmgsbemgaponny.. ° A«. 9920.
I. J f;r'87lhunSf: p ■="' " ^■''"'"^' '' ™r "f;,^;
'TO IT 1° .1, Ans. 1479.
ton f "''' ' ^^^ ^^"^ ^^ ^^***^^ ^°«<^ ''^ ^25 a
'TO 71? 1 /. 7l7w, 012;')
will 119 n^ ^T f /°^ *^^°S cost 4 pence, how much
80 llZ " '"'* • • ^^" 448 pence.
80 Row njany pence in 100 pieces of coin, each of
which 18 worth 57 pence ? a,,, r'inn
CI TT^,„ f^^^^ jins. 5700 pence.
t«inL RQ ^^^ypUons in 264 hogsheads, each con
taining 63 gallons .? ° L,, Tp,'"
82. If the interest of ^1 be £005, how much wi5i
be the intarest of ^6376 ? ' Ans £l 8 8
cosll" "^ ''''^ """^'"^^ '°'* ^^'^^' ""^^^ ^^^^ 973 such
84. It has been computed that tho gold, silver and
brass expended in building the temple%f So omon at
Jerusalem amounted in value to ^£6904822500 of our
money ; how many pence are there in this sum oZ
Ts'Stir"'' . 1 ^165VV57SC
85 I he followmg are the lengths of a degree of the
?em •1?):86'6' 't"?"^ ^"""^'^ 604802\ThoLin
. n ' ,^^^^^^ m I»<iia ; 607594 in France • fiOS^fifi
m England; and 609524 in Lapland. 6 feet beln' a
st^Yt'^ r"^ ^''' ^^ ^^^^ «f ^' above. P 1?.;
362b8l2 m Peru; 3629196 in India; 3645564 in
France ;r^650_l 96 in England ; and 3657144 in Lapland
86 The width of the Menai bridge between ?he
points of suspension is 560 feet ; and th? weigtrbeLL
these two points 489 tons. 12 inches bein/a foot and
2240 pounds a ton, how many inches in°the fomTr
and pounds m the latter ? loimer,
87 Th..n ^"1' ^^2^.i?«^es, and 1095360 pounds.
87 There are two minims to a semibreve • two
crotchets to a minim ; two quavers to a ciSeV wo
semiquavers to a quaver : and two demisem quavcrslo
I rerSre^P ^^^ demisemiquavers i ~^
Ans. 221
M'
M ULTI PLICA n ON.
77
88. 32,000 seeds have been counted in a single poppy •
how many would be found in 297 of these ? Ans. y50400o!
89 9,344,000 eggs have been found in a single cod
Lsh J how many would there be in 35 such ?
^^ „„ , Ans. 327040000.
^ 65 When the pupil is ftimiliar with multiplication,
m workmg, for instance, the following example,
897351, multiplicand.
4, multiplier.
3589404, product.
He should say :— 4 (the product of 4 and 1), 20 (the pro
duct of 4 and 5), 14 (the product of 4 and 3 plus 2, to be
earned), 29, 38, 35; at the same time putting down
the units, and carryin^; the tens of each.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
1. What is multiplication .? [24].
2. What are the multiplicand, multiplier, and nro
duct.? [24]. ^ ' ^
3. What are factors, and submultiples } [24] .
^ 4. What is tlie difference between prime and compo
site numbers [25] ; and between those which are prime
and those which are composite to each other ? [27] .
5. What is the measure, aliquot part, or submultiplo
of a quantity ? [26] .
6. What is a multiple .' [29].
7. What is a common measure ? [27T .
8. What is meant by the greatest common measure >
[28] .
9. What is a c<9wmo% multiple .? [30].
10. What is meant by the kast common multinle >
[30]. ^ •
11. What are equimultiples .? [31].
12. Does the use of the multiplication table prevent
multiplication from being a species of addition .? [33].
13. Who first constructed this table ? [33].
14. What is the sign used for multiplication ? [34].
15. How are quantities under the vinculum affecteJ!*
by the sign of multiplication .? [34] .
16. Show that quantities connected by the sign o/
multiplication may be read in any order ? [35] .
WL^ai
78
iiirisioN.
,0*';^',"'^ ""■■ '""llilJiw exceeds 12 > ran '
cccoAs I2t [Lt "'''' "'"" °»'^ *» nluUiplicHud
*; J°' '^^^^''"■« tl"= rules when the mnltipUcand mnl
' a^'^Tr; ""''.?""" "W'"''^' ^ decimals ?[43;&Ti:
2i' F."^i' °'""'P"''f<'? P^o^^i •' [42 and 53].
wtdgn?.Sl' ITsr""^'' ""^°"' ^"pp»g »
n»nfbo?„7del:rpLt¥^r5sV' '" ''"^ " '"^^
^ J8. How may we multiply by any number of Jnes >
[esf: ^""^ '"" °"'"'P'i''^«en «ry briefly performed ?
SIMPLE DIVISION.
orde'eminatl""""'' "^ ^PJ*""*' ^■" "»' o^"^
calMTlTJ?'^""'™ '" *■"> °"' ''"w oft™ one number
called the divtsor, is mnlaineil in, or can *, /^/J;,T '
another, termed the divi,i,^,l ■ .1 ^ i ™'«"/™«
/».//» is call d tt S;„T DKr™'f^P''"^T«
u« t» tell, if a quantity be dWded into a cL?" '""f '
"'Xi^^'d^""' ^"^ "^ ti'lCt^ofrh" """'"
wnen the divisor is not contained in the divid^n.?
p.ocea, would be required to'dLTrer^SyTub*
vivitioa.
70
trading it~liow ofton 7 is contairiod in 8063495724
jv^i.lo, a.s wo_«h.ll liud, the same thing can bo effoctcd
by dicmmi, m less than a minute.
08. Division is expressed by f, pLvced between the
.v.dend and divisor; or by j.utting^he divisor und J
the dividend, with a separating line between :— thus
643=2, or=.2 (road (3 divided by 3 is equal to 2)
means, that if 6 is divided by 3, the quotient will be 2.
<)9. When a quantity under the vinculum is to ).c
divided, we must, on removing the vinculum, put the
divisor under each of the terms connected by the .i^^u
of addition, or subtraction, otherwise the value of wliat
was to be divided will bo changed ;— thus 5T6^^3=:r
6 7 ' ~
!.____. f^^. ^^j j^^ j^^^ ^^.^..^^^^ ^1^^ ^j^^^^ ^^^^^^^
vvc divide all its parts.
_ The line placed between the dividend and divisor occa
Monally assumes tlie place of a vinculum ; and there
ore, when the quantity to be divided is subtractive, it
\uli sometimes be necessary to change ^'  *
already directed [16]:— thus + ^^""^
the signs— as
6 + 13—3:
but
27
3
^5— 6 + 9 ^ 27—15 + 6
3
2
9
2
For when, aa
in these cases, «// the terms are put under the vinculum
"thf : '' v^f '^^? '"^^^^^*^''^ ^^S"^ ^'^ concern S
IS the same as if the vinculum were iSmoved altoether •
and then the signs should be changed i../; .^a tJ
what they must be considered to have been S tl e
vmculum was alfiv^ed [16]. ^ ^
When quantities connected by the sjcrn of multinlien
tion are to be divided, dividing Iny one° of trS.s'
.nil be the_same as dividing the product ; thus, 5X10 X
2o+5=  X lOX 25 ; for each is equal to 250.
To Divide QuanlUits.
70. When the divisor does not exceed 12, nor the
dividend 12 times the divisor '
80
DIVISIOIf.
B.;le.~I imd by tlio mnltiplicntion taMo tho
groutost number which, multiplied by the divisor wlU
give a product that doo.s not exceed L diviS' 7\^
will be the quotient required '
number td'fh/r^" '^'' t^^''f *^« P^«^^«* «^ ^^"'^
anv w tlw i • ''^''"' ', ''^'"« '^'^^^^ *^^" remainder, if
any, with the divisor uuder it, and a lino between them
aro^S T '''^ ^^/""Itiplication tablollllo times G
The total quotient is9+,or9^; that is, ^=9^
^ji :^:S^t^XS^;rt^"^or, w^ean^^ct it
ber'of^i^rThrcivTsoxLnt;\i^ *?« f** 
is, tlie greatest mull pie of 6 w], oh •i^'''"V'^^ dividend; that
ber to°be divided The mnlTinH^^^ not exceed tho num
ducts of any two m^mhL '^ '^"^ ^^^'^^^ ''^'"^^■^ the pro
therefore iJirlblesu" to obt^i"^^!^^ ' "'^' ''''''^' ^^' '^"'^
must not exceed the dhiSend"^^ ^''^^^"^■«' ^•"'*
leave a number equal to or 'olf' T° «".''t^''^«ted from it.
hardly necessary loremrk t at .7 * i'""' '^'' ^^^^'•^«^ ^^ is
been subtracter! .4 ofTon 1.' '.t k'*" ?"'''"'' ^^"^'^ "«t have
number equal to or ffreatel tlfan u'' ^''? i''« ^^'^^'^^"^ ^^ ^
quotient answer tleon^^Hnn / '^ r '''■'' ^'^^^ ' "'^^ ^^0^1'^ ti.e
taken from the dividend °' ''"' "^'"'^ ^^'^ '^^^^^^^^ ««^^IJ bo
anf^SenTf(4;r'JreSrnVL^ T'^^V ^^ ^'^ ^^«
remainder, what it i When^j. Je H';^ *'"i'' ^' '^"^
Jeahty suppose tho dividend div eel iito IT'^^'f'' ""' '"
these IS equal to the product nfThi "^^ P'''^"*^ ^ «"« f>t'
pie given, f =:^i±4=,l4 4 4
72. When the divLsor does not exceed 12 bui M,«
dividend exceeds 12 times the divisor ' ''
DIVI.SION.
tablo tlio
li visor, will
idend: this
luct of this
mainder, if
yeen them.
1 58; or, in
)y G.
10 times G
Teforo, doea
ible, that 9
tly 6 is con
9 quotient;
given num
4
an effect it
ateat num
dend; that
I tho mini
's the pro
Is 12; iitid
luire; this
id from it,
sor. It is
1 not liave
idend if a
would tlie
■ could bo
le divisor
be any
er, we in
s ; one of
lent— and
i between
^s, hy the
lie cxtiia,
bui the
81
liiiHBssias
r
10
ino
mg remainder, when there is one ind de /r/ '^I
contain the divisor consido,. .V + ' ^"r"^"^) ^^^«« "ot
the next lower and ^d, I ^ '''''' ^^ "'^'"^ ^^'
the dociuml point. ' ^^'''' removed from
"Jis^et;:rit,"rit!t:; ^'^^ ^t^^^ *'
[70],vith the div sor u nd l 7 i ^'^''"''^^ ^^'""'^^'^^
l'otv.oen tliem ; o ^writ nt tl!' d" ' f'^"'"'"^^' ^^"«
quotient, proceU with t o^di! ""i^ ^^''"* ^" *''^
vonKundor'ten 1?^! t t nt^^'th'e ttt'l""'"' r''
mmathm ; proceed tliua Z^7fh • ^ ^''''''^'' ^*^"'^
nntil it is so tr flin .7h. T "'? '' "^ remainder, or
iaconvenfence ° '^"^ ^' "'^^Slected without
73. Ex.MrLE.What is the quotient of 04450^7 ^
Divisor 7)G445G dividend. ' '
1)208 quotient.
i« greater thin GO t mm ?1 '' •'^'' '1* '^^^'"'''"'^^' ^^'^'^^''^
to he nut mle J T: \^'T '"' therefore, no di^it
o;>evei, put a cyp icr m that place, since no digit
B2
DivisroN.
l!
drnfli4 '^ ■ • HI11L3 ^ JiiinUrods nvo (iT.ir.n,r i i i
the
go
tens, which
tens
divid(
in«
phice of the quotioi.L, juiu cue
quotient is fu,i„d to bo 9208 eaetly ; H.at is, tt^^ 92O8.
b/fi J ^""'"'^ 2.What is tl,e quotient of 72208, divided
0)73208
proceed with the division as followa ^ '' "'"^
0)73208
' ■ ., . , i22rf333,&c.
Lonsiderinoj the 2 units inPt f„ xi
dend, as 20 Tenths, rpelceivetZ a" ".f ' '^ ^''^ ^^^^^
three tenths times and I^.tn if •''''" «'' '"t" t^^o'^
6 (=0 times 3 emhTf^t^f ' I ;f^^ ^ ^^"^^'^ ^^'''^
into .cThund^edths Vl:^^^ ^^^ wn;^^
Ilin dooiiutil
euMo, i»ro(Iiu!n
of tlK.usatiil.s
^iiiu.l» iilrciidy
'• "so" into
usiind times;
I is Icsa tliau
oos not Icavo
it is not too
I)laco (if tlio
ng adcloj to
•^ thdiisaiul)
iivo 14 Imii
il.s, and Icavo
3tly 14 Jmn
da' ])Iai!o of
7 will not
Jnes 7 are 7
Ji'in;; tho 5
units of tho
tiniOH, loav
in the tens'
urea fiii'tJior
ni nation of
phor. Tho
!»o required
=9208.
28, divided
■ after tho
>i" wo may
' Hio divi
into thoui
itlis times
t <> in the
onths ro
(i will go
e 2 lum
DIVISION.
83
denominations of f]/c qu.)tient; wo m v^ 1 o ■ ""'r''^'«
put dowa m the quotient us nutn^ S^. s w ^.'^.'7^
liual remainder so small, that it mtiy be m'glecTid. '
75. Example 3.— Divide 473G5 by 12
12)47305 '
* 3U4708, &o.
In thia oxampie, tho one unit loft Caftcr obtaininn fKn 7 •
ntr'i^ tr ^'r. r^^i as io to^rsrit
StZths'';;;;x e^^^^^^ "^^^^'"« to bo set down ;
ine renins place ol tho quotient— except a CYphcr to kr.pr.
tho following digits in their proper places 'Fm in ? !i ^
are by consequJnco to bo coLiXro. ^^^
12 will go int^ 100 l^undr^dZ 8 idr tl '^^
toliSuYtt iSn^^ ''' '' ^« m whTn^rdosiro '
Example.— Divide 8 by 5.
«^5 = l'Sorl37, &o.
76 When tho pupil fully understands tho real deno
n^inations of the dividend and quotient, he may proceed
for example, with the following "^ pioceeu,
5)40325
In this manner :— 5 will not go into 4 5 I'nfn da a r
find 1 over Cthe 40 bpi'nn. r>f Jv i • . ^"' ^ *i»ios
produced it) 5Tnto 13%w ^"' the den<imination which
times and 2!,ver' 5 in l oJTtir:^ ^ r^' ^ ?"^° ^2, 6
' ""^^^ ^"^ ^^^ remainder.
ligil. of the quotL^aro a/eo Str'lV' '' Ti"""" »'' «>«
.ub.™t.. rro„ t.4^"^nf istsi;™ :i,'ir.ii' riL°
84
nivisiiotv.
Thni, }f T) gooji
tlio quotient (544.k5l7 w« '?„ ^,/^' ';• '", '""'"'f?' ["•' «xa»iplo,
cnvi.Iond Huitc 1 to M, roCs"'? , '• "• '^'^'''••••^»T. to render tl.o
>vlule, at tI.o «amo t me/wo e vo i s v^h'"' '''"■'•'"•• i'''^ '"'"•'"•
coruort • *^" ^*''^^ "** value unohangod; It bo
Thousands. Hnr„lreJ«. Tuns ' ir •,
E«wh j.art being divided hv 7 f l,n hr ^'^' (^^^^f^^)'
dividend, with tifeir ^iotiv'o q'uotonln^'^Jirbr''^^'^^ "' ^'^^^
and the nuestion is Zf./v *: '^ quotient in a lower;
dividend~it3di?Lint de onf/n'V '" i''''''' ^''^ «" '"to tl'O
venicnt way \Vo can n T^ T ^"'"» taken in a,;y con
Hhall have t^o add L t e LerdLonX.';;' "^ '' 'I'' '''^' ^^°
With the higher. '"^ ^''^^^^ denominations, unlesa we begin
th^x^n5;Lf"r;f S divide:;!; 'Y r''^?'  p^* "^o
it belong,, to that demmdnatiZ "!;';'''' ^''''^'''''\ '^' ^««'^»««
of time "(indicated by S of'thn. expresses wliat rumber
tan bo taken from the coSInn'^"'^ "lenonunation) the <livi.sop
thus tlie tons of the onot^^n?^ "" T"'^ ""^ *''« 'Jividend:
the divisor can be t ken fro^lT'''^; liow many tens of times
hnndreds of ti^e mio ient tZ "" *,'"' ?^ *''« dividend; tho
6e taken from tlie TuXds &c '"^ '''''^'''^' '^ '''^'' ''^ '^^
m^:ih::;it^;;j;^i;So7^;, l^f belong, to the total re
lower denomination t w i tti ' m •* "'i^''" considered as of a
He Asorv OK V Ve a Jo f o 1 ''l""" ^"'':^"^*^^ ^" *''« quotient,
the highest deLm nation canablo n^ ■^'' ^"''"^'""^^^^ ^« «^
tliougli it may not cont^i n E J?„?^ ^^^"^^ i^ quotient; and
press«. by a di<r it of one don., • I'"'" 5' '''''^^^'' ''^ times ex
Lnbc., ot^im J^/p^oTed'ryTn^Ua^ «^"^' ^^ 
t2!J:reSnXrtheS[t^5eS^S'"^ oach.produot, is the
*) mucjj of it as is neoessirv f^ '/"'^ ^^ ^^''"3 down" only
iooking for a d L t ?n ?£ JmnX';;' Pr''"^°^J^°* 'Th^^' in
^m not be ncc^essary tota^e Sy;^^^^^ quotient, it
Of the dividend ; since hev canm f « n .""?, *^'' *'"''' °^ ""^ts
<lre.s of times the aivi.iJ?„^;rfaS f^oI^thrriS.'^'^*
DIVISION.
85
Acyvher mnst bo n.Mc.l [Sec. I. 28], when It ia rcquiml
tlio orwe. except it oinoM botwcou thoiu un.l me .leciuml pint.
Hr:AsoN <.K VI.Wo .....y continuo ti.o process of division.
If wo plouso, a.s o„ff as it ih pusible to obtain .,uotio»ts of ani
icnon.m.uoa. Q.iotients will l.o po<luce.l although th ore are
nu lung.r any «.gn.nount figures L the dividend, to which wo
can add tho succcHHivo roniaindora.
78. Thfi BiiialW' the divisor the larger tho (motiont
lor, the smaller the parts of a given quantity, tho groatr
tlum nuuiber will bo ; bt.t is tho least po.ssiblo tlivi
sor, and therefore any quantity divided ))y will <rivc tho
larg(.st possible (^uoticntwhich in infinity. "Hence
though atiy quantity multiplied ' by is equal to 0, any
number divided by is e.pia! to an infinite number.
It appears strange, but yet it is true, that=: ; for
each is equal to tho gre.nlr.st pns.siblo number, and one,
thcrefc 3, cannot be greater than anotherthe appa
rent contradiction arises from our being unable to form
a true conception of an infinll.c (luantity. It is neces.^ary
n.m^ •^•'Tf '^'*\f'^"^ ^ ''' ^^''' ^'^«^^ "^dicates i
quj tKy^inhnitely small, rather than absolutely nothin^r
7J lo 2>roic Dlcis}^n.~lSlnmiy\y the quotient fv
the divisor ; the product should bo equal to the divi'
Uend, minus tho remainder, if there is one
I'or, tlio dividcn.I, exclusive of t!ic ronaiuder, contains tho
Ir'^t' rdivl.'": '• ""rr '""''r'''' '^^ ^^c quotient ";?.«!
eoml nf r ',''i^''^'^'" ^'"'^ ""'"^'^'r of times, a quantity
U i 1. tvH tirirnTr • "",'"' '''« rnaindcr, will be p^Uucelf
it luiiows, tiiat adding tho remainder to tlie product of the
divisor and quotient ahould ^ve the dividend. ^ ^
EXAMTLK 1.
T708
n i.t i. 'L'^^32 ,_
rrovo that — ^=1708
4
PuooF. 1708, quotient.
4, diviKur.
sov and quotient, equal to the^dividcrlf'^' ^^'"'^"'^ "^' '^'^■"
ExARipi.E 2..l>r<,vo tliat ~"^^^ ioom ^
Proof. ' , '
rnooF.
or 122^4
122^1
7
**^^*^^=='iivMcu>l
"limn:., tliediuniii V
7
i: 2
86
DIVISION.
2)78345
EXERCISES.
. <2)
8)91234
(3)
3)67869
(4)
9)71234
4)96707
6)970763
(6)
10)134667
12)876967
(7)
6)767456
7)891023
(8)
11)37087
(12)
9)763457
SO. iVhe>i the dividend, divisor nr hnth
w hen the dividend contains cyphers—
68 7 times, it will bo ntrt fiRrin / "^P^^^' if 8 w^l go into
than 66) 100 tiSS ^orftLtTtiL^s^ir^oV.TJr^ ^•'^^^*
i^AMPLE l.What; is the quotient of 568000^4 i
4 14 J, therefore — —. = 142000.
E^mPLE 2.What IB the q, ti^nt of 40G0000.5 ?
*; SI 2, therefore —y— =812000 [Sec. L 39.].
81. When the divisor contains cyphers—
can be taken fr m it 100 tfmes^le^rSen. ''' ^^'^ '""^' ^
ExAMPi.E..Wliat is the quotient of ^ I
ro °00
  58
■g=/ ; therefore ~= 07.
800"
DIVISION.
87
(4)
171234
(B)
37087
12)
33457
■ contairi
re apjtli,
them.
removo
2rc liavo
) be tlie
e Uiviaor
' go into
d greater
; 2
39.].
we the
cyphers
times it
'; 6 can
times 6
82. If both dindenrl and divisor contain cyphers—
RuLE.DA'id^ as if there were none, and move the
quotient a number of places equal to 'the dfiL'enoe
t^Hr'''?^""^?'' '^ ^^P^^^*^ "^ *he two given quan!
t ties rif the cyphers in the dividend excefd thole hi
the dmsor, move to the left; if the cyphers i^ the
divisor exceed those in the dividend, move^to Te T^ght
ExAMPLEa.
(i)
7)63
9
(2)
7)6300
(3)
70)63
" 09
70)6300 700)630
(6)
700)6300
eoo 09 —90 —0^ 9
of Pvnit^l^ ^^^"WK the difference between the numbers
83. If there are decimals in the dividend—
KuLE.— Divide as if there were none, and move the
quotient so many places to the right as there are deoi
The smaller the dividend, the less the quotient.
ii^XAMPLE.— What is the quotient of 04858 ^
48 048
g— 0, therefore ^==006.
84. If there are decimals in the divisor—
KuLE.— Divide as if there were none, and move the
quotient so many places to the left as there are deci!
The smaller the divisor, the greater the.quotient.
JiXAMPLE.—What is the quotient of 54i006 '
54 54 ' ■
g=9, thererore;^==9000.
visor"^^ *^'''^ """^ '^''*'"'^^' '"^ ' '*^* ^^^^^^^^ «n*^ <Ji
Kui.E.— Divide as if there were none, and move the
quotient a number of places equal to the difference
M
68
DIVISION.
between the numbers of deeiraab in the two given quan
t.t.e« :_.f the deciraab in the dividend exeeed thor?n
the divisor, move to the right : if the decimals in tim
divisor exceed those in the dividend, move to the left
Examples.
(1)
6)45
9
(2)
6) '45
•09
(3)
•05)45
900
(4)
•5) 045
•09
(5)
•005J_450
90000
(6)
•05) 45
900
■ *],f^^''^;'~~?''^'^^ ^^ ^ *^^^« ^«r« i^either, and move
the quotient a number of places to the left, equdlo
the number of both cyphers and decimals. ^
Example.— What is the quotient of 270f03 ;
^=9, therefore, 270^03=9000.
in ?he d^koTl'" ^'""^'^' " '^' ^^^^^^^°^' ^ «^^«
Rule Divide as if there were neither, and move
the quotient a number of places to the ri4t eauS to
the number of both cyphers and decimals. " ^
^Zi^^Z^S^^'"'^' '^' « «^P^  the
E.YAMPLE.~Whaf ia the quotient of 18^20 ?
^ = 9, therefoe ■l = 009.
20
The rules which relate to the management of cyphers
and decimals, m multiplication and iS division/hou4
iumerouswill be very easily remembered, if the pupil
. " ■' — """^ ^d^fcc. tu yu ine cuesi v) ciihor
;iven quan
id those in
nals in thu
the left.
I move the
ivisor move
ogether, the
eir separate
(6)
'05) 45
900
and deci
md move
equal to
a la in the
Icyph
ers
ad move
equal to
ra in the
cyphers
though
he pupil
iihtir
(13)
8)10000
(14)
11)10000
DIVISION.
EXERCISKS.
(15)
3)70170
(16)
6)68630
89
(17)
20)36623
(18)
3000)47865
(1^)
40)56020
(20)
80)75686
(21)
12)63076
(22)
10) 08766
(23)
•07)64268
(24)
•09)57368
(25)
•0005)60300
(26)
700) 03576
(27)
•008)57362
(28)
400)63700
(29)
110)97634
88. When the divisor exceeds 12 .
The process used is called Ions; division • thif i<? wa
perform the multiplications, subtlaclLrr&c , nJl
and not, as before, merely in the mind. ' This will be
imderstood better, by applying the method of longdivi!
Tat r ihanTr^ '• ^" ^^^'^^^~'^ divisor noticing
gi ater tlian 12 — it is unnecessary. °
Short Division :
8)6763472
720134
the same by
Long Division.
8)6763472(7204:]4
56
16
16
>
34
32
27
24
~82
32
dil?renM,aTof X'f 'r^"^^;^!^ *^« ^^ ^'J the
uuiuent paits ot tJie quotient, and in eacli case ,ei \\mo,x
90
DIVISION
the rroducf mLlmct it from «,« corresponding portion of
the <1 vKlond, write tlie romaindor, and irinn ffJnth^^,
quuod digu, ,>f the dividend, Ali this 3 beTne when"
dent' .StH^i^te™' *" '^'^ '''' "' ""^ *""
III. Fidd the smallest number of dibits at ih<^ 7pff
and set down, underneath, the remainder, if there is
any. Ihe digit by which we have multiplied the divisor
IS to be placed in the quotient. ^
. T* J? *.^''' remainder just mentioned add, or, as it is
said " bring down" so many of the next' digit or
cyphers as the case may be) of the dividend, as are
required to make a quantity not less than the divisor
and for every digit or cypher of the dividend thus
brought down ^c.^^ .,^, add a cypher after the digit
last placed m the quotient. ^
VI. Find out, and set down in the Quotient iht>
nnmUr of times the divisor is contained if^hTs qUn!
tity ; and then subtract from the latter the product of
tlie divisor and the digit of the quotient just set down.
J^roceod A^.th the resulting remainder, and with all that
succeed, as with the last.
A ^?'}l *^®/^ '^ "^ remainder, after the units of the
dividend have been " brought down" and divided, either
place It into the quotient with the divisor under it, and
a separating Ime between them [70] ; or, putting the
decimal point in the quotientand adding to the re
mainder as many cyphers as will make it at least equal
to the divisor, and to the quotient as many cyphers
mi7ius 071^ as there have been cyphers added to the
remamder— proceed with the division.
DIVISION.
portion of
^wi the re
done when
lid be too
the divi
e for the
■ the left
quantity
lem, the
contain ;
there is
e divisor
', as it is
igita (or
, as are
divisor ;
Qd thus
he digit
mt, the
s quan
duct of
t down,
all that
of the
, either
it, and
ing the
the re
t equal
lyphers
to the
t)0. ExAMPLK 1.— Divide 78325826 by 82.
91
82)78325826(955193
738
452
410
425
410
158
82
762
738
246
246
fnt?7« V '' ^1 ''l*^ ^ \ ^°' i"*^ ' ^ 5 ^»<^ it ^^i" go 9 times
tV T^ ''i'',^^ P"* i" *^^e quotient.
xvni L T''^*,^*' ^^Shor denominations in the quotient
if the DunTl ns hT '''' 'T''"^'' ^'^ P^'^P«^ t« a«certo,in,
ff titrtr!;h7ch th^e^roit" ^^^^^^^^^^^ '^'^'^ *^ ^^
6 thnes'"! t'"^^* ^.'^'"'..^^ ^^^« 4^ i^^to^vhich 83 ^e
LiiB uivihoi tiom 4&J, which leaves 42 as remainrlpr 49
with 5, the next digit of the dividend, makes Sr^nowhith
um me quotient. Ihe last remainder, 15, with 8 tliP n^vf
digit of the dividend, makes 158, into whcr82 toi once^
leaving 76 as remainder:—! is to be nntln tK^ g?es once,
rlToK^l ^ ^""Stit down expressed mut*
t , ij
Therefore
78325826
82"
=955193.
02
DIVISION.
Sample 2.I)ivido G4212S4 by G42
042)(;42i284('T^( 02
042 ^
1284
12S4
ing tr'thrncx?d?'^' r??, ^^^r^"^ ^. nrl,..
1. TJh) next di^ of 't e 1 !n ' T^'" ''' '^t'''^"^' ••^''^«^ *!'<^
«o digit in tJic^ not on h vhS "' '''' '^^'"'^ ^^''^^' g'^^'^
another cypher: and for sin l r vf ' '^""^'^^l^^^tlj^ ^vo put
down the next •X/ihn?''v ''."'' '*'"'^'^" i» hibJivr
gives^io romainderUve put 2 in'fT "'' f^'^''^"^' ''''''' ^"^
91 Whm. +1.nv^ • "^P"*^ J" <^^"c quotient.
division, adding deoimaS SesfoTl '^''^ T'^^ ^""*^""« "'«
o uLcimai places to the quotient, as follows—
ExAMPM 3.— Divide 79G347 by 847.
847)79G347((i4010,'&c.
3404
3388
convenient to have two on nSS Zi ■ '""^^ *''* ^'^ '' ^"^^'^
!i;fl
DIVISION. 93
6425x 54
^± ViTsT&c.
102
64^
485
432
53, &c
Rkason of n.This, also, is only a matter of convenience
Rkason OF III.A smaller part of the dividenl woul "ivo
no digit m the quotient, and a larger would give more than
Keason of IV.Since the numbers to be multiplied, and
the products to be subtracted, arc considerable, it s not so
convenient as m short division, to perform themultiplicXns
and subtrac ions mentally. The i^ile directs us to set c bwu
onlTLitfe" "" ^"^^■^^^^' '^^^"«« ''' ^^^*  *^ 
liFAsoN OF V.One digit of the dividend brought down
would make the quantity to be divided one denomination lowe?
than the preceding, and the resulting digit of the ouEt
also one denomination lower. But if we are obliged tTbrfn 
down two digits, the quantity to be divided is fwo denoiS?
nations lower and consequently the resulting digit of tlie ot^o
tient is ^^/»fl_ denominations lower than the preccdinoiwhich
uro^a'c^vnliet^^'/.r*'*"" ^'^" '■ 281 is exp^ess'^St;
using a cypher In the same way, bringing down three
S't' '/ the dj^j^^^^^ ^^^j^^^^ ^^^ denomination thrle places
lower^£n the hZ^ '' "f ''''''''''' ""'^^^ clenominSs
lower tnan the last— two cyphers must then be used Thn
T.^TT'^'V'''^^? ^"'^ an/number of characteis whether
significant or otherwise, brouht down to any remainder
J^^l^fl ""^ ^^.^^« ^"bt^"'^* the products of the different
parts of the quotient and the divisor (those different Ss of
are ?ounS th^„'/°^ ^^^ '^7'' «"«««««ely accorSg^aB they
are tound), that we may discover what the remainder is from
w W J'A'''' *', "^Pf '* *^^ "^^* P^''*^*^" «f the quotient FroS
no 3ecTmaYs?nT1? '^ ^IP' '' ''■''''''''' *'^<^*' if there ar^
noaecimaism the divisor, the quotient fifmre will alwiv^ hi
It is proper to put a dot over each dimt of tlie divi
dend, as we bring it down ; this will prevent our Ltei
tmg any one, or bringing it down twice. ^
94. When there are cyphers, decimals, or both, tho
94
DlVIiJJOX.
9o. lo prove the Ihmion.—mxMMy the ouoticnt
by the divisor ; tLe product .should be iL\ to the div 
dcnd, mums the remainder, if there is anV '791
^^.^io^Fovo It by tho method of "casting out tho
KuLE.— Ciist tho nines out of the divisor, and tho
quotient ; multiply the remai.ider.s, and cast tlirni it
fiH„u tiieir product :that which L now left oi^ to
out of tlie dividend minus the remainder obkined from
the process of division. «"ucu uom
Example.— Prove that ~J!~= 1 181 .3
Con^ider^daBa'^
;, ,. ^ '^'*^^7762 = 037/4. To try if this bo true,
Casting the nines from 1181, the remainder is 2. ), .
.. ". " i^'Oin 54, „ jn 2x0 =
tasting the nines from G3774, tho remainder is . .0
The two remainders are equal, both beinoQ henco tl.n
multiplication is to be presumed right, ancf «Zi Iv
the process of division which suppos "s it. ^^'"'^^'Ititntly.
The division involves an example of multiDlicatinn • sJn^.*
the product of tlie divisor and q,loticnt oi^ghfto be cnuaMo
the dividend minus the re.uainder [7'J]. lieLe in mot^nc?
!idiri^^;^zirir^'  ^'^y e.pij.;rc£;nf
EXKRCISiCS.
(30)
24)7054
(31)
15)0783
318f3
(32)
10)5074
452^,
(33)
.* 17)4075
35410
275
(34)
18)7831
(35)
10)5977
435A
(30)
21)G78r
(38)
23)707500
3141^
(39)
390)5807
(37)
22)9707
323
443i
33309^5
(40) ,
1400)0707000
148897
40353425
ic quotient
to the divi
g out; tho
•, and tho
tlip nines
1 ought to
tho nines
iued fioni
3becomos
truo,
2x0 =
.
henco the
iso(;[uontly,
On ; since
i equal to
a proving
[54], we
(33)
'17)4075
"275
(37)
2)9707
'ml
000 .
3353425
4
(41)
250)77670700
303424 •00y4
(44)
6426 )123 •705 86
2 2803"
DIVISION.
(42)
671^^42
•002
96
(48)
•163 ) 8297 49
54232 '
(45)
14 86 )269 0625
1875
(46)
•0087 ) 655
150000
In example 40— and some of those which follow— after
obtaming as many decimal places in the quS as I'o
deemed necessary it wiU be more accurate to cons der t^ic
haTf'o?ftT Z?"u *^f ^^\r^r («i^«e it is more than one
halt of It), and add unity to the last digit of the quotient.
CONTRACTIONS IN DIVISION.
96. We raay abbreviato the process of division when
rflf f^%«^ajy decimals, by cutting off a digit to the
right hand of the divisor, at each new di4 of the
quotient; remembering to carry what would have been
«n tTff th^ the mdtiplication of the figure neglected!!!
unity if this multiplication would have produced more
25r&c' T59T ^^ ' ^ '^ ""'"' *^''' ^^' "' ^''' *^""
Example.— Divide 754337385 by 61347.
fli oA?!n.TLT^'''^ Contracted M^od.
61347)75^33 7385(12296 61347)754337385(12296
C1347
14086:7
12269 4
1817
33
I226j94
590J398
652 123
38
36
2755
8082
i46730
61347
14086
12269
1817
1227
"590
552
li
37
DC
I'lVlSiOX.
;i'untly. tho portions of the dividcna from vMch th ^ T'n
liivo been Hubtracted. "What hIiouI.1 i.uvn h •^ .^ '"''^
the multiplication of ti.e digi nSteHin^^^^^^ '■"'"
97. Wo may avaU ouiselves, in diviaion, of oonfrJ
vances very similar to those 'used iu m;Uil>lS;,n
To divide by a composite number—
lluLE— Divide successively by its factors.
Example. Divide 98 by 49. 49=7x7
7)98
7)14
"2=987x7,or49.
98 If the divisor is not a composite number wp
canno,as in multiplication, abbreLte' tC process
oxcept It IS a quantity which is but little less^than a
c^edTthe """^ f"^' *^" ^"^*^^"'  the _d
Tn.^ A ?''''"' i'^'"^"' «^^ ^i^'i^e the sum by the
precedmg divisor. Proceed thus, adding to thTrcLi '
der in each case so many times 'the foregoing .uoHcn
or i s'nffiT^f '^'^^^^ '^' Siven divisor untilV "xlc^,
i?^. atcTlLT/l^P ''""''^*^'^\*^ *^^ exact quotien
IS obtainedthe /c5^ divisor must be the given, and not
the assumed one. The last remainder will be the ti ue
tilVlHlON.
fff.
K.VAM,.r,K.~I)ivido l)87(5r.;342r, by 998
9870G;j.,425=t)S70(i342r)^10()u"
i97r>.jr.l=ys7lTo33^2qp425^U)00
4.JU1=.IU752^75T41000. ■'
07..0U0=4x2:701>]00()
. 0()l4)4()=7x2.flT^1000
OU00..420=:.ur^2:p4_^100().
00004,,0208=aT^2+^^908
icnutiS^" '"' ^^^^^««*  «^004, ^a 0208 is the lu«t
r 987G03
1975
all tlie quotients are • '^ »
07
OOl
I 00004 V '
The true quotient is 9890427lO4 », ^''•'^'**
And the true ren.uindor 0.0908 ' Z u 7" "'' '^' .l"""""''
Uj 1 ^ u i'u«, or tlie iast reniaindew
or the part qf it ju,t UiSd lin \ ^»«'n t^'e dividcul,
tlio third lino 4701 ,nV J^''\ ^I"otient. Thus in
as quotientrand ^Tof ^itf s'jl'rt' ^ ]T^ ^ ^^^^^
701 us ren/ainde/ 47 wo ] ^'^^'^^^that is,
to the Icl/of tie decinn \^/' ;>^^^P.Vin^ four places, al
pv units as quj^;^ r^'al^^tlf^^ ^ ^^««'
liue), one is a decimal i.!.. *! ^ ^ ^^" *^^ »ext
tenths; and in OlOJO InJt y \ ■ ^' "' ""= '"''ier
four piaoos are doo m iT f j '^'r'""" '™ "»' "^ 'he
dreJths, &o. ""'"»■"''. "« quotient must be huu
plied, and\he .:,,'';:;".";' '?"»/' Vtient multi^
oddcd, be]
onix
o
•oiiiaindcr to wliich the product
t.ts
IS to be
98
DIVISION.
^
47
4H,
4!),
60,
CI.
62.
ca.
ct,
Go.
(SQ.
C7.
68.
61).
00.
v(il.
(52.
(53.
(54.
G5.
m.
G7.
08.
EXKHCtNKa.
, r)0789f.74l==70*7;«
47H!M>7M)71=4l>;;JL'f.
1)77070 +'17(50(3=20 ?hZI.
607807 ^8 12=074 •"!•
78(57074 f071 2=816 W.
JK)7O7()Of457O()O=(J7l03.
07051 68 ^7894=8r)7.
tt7470^.;{l)00=173.
0OO0Of47()0O=l 4490.
70707 ^ 40700=1 8802.
(51M692T704;^24=8.
9070744^9] 0070=1 • 0.'J29.
740070000 ^741000=998 7449.
94 10(507 1 1 1 f45078=200043 • 1 1 32.
45407(5000^400100=1 1 349003.
737(547(57(57 Ha46(570=:2i::;39 049.
47 ;6782975^20• 175=1 8177.
47 (355^4 5=10 59.
76098^7G7301 2=9 • 800.
75 3470+8829=190 7798.
0'l+70345=00000131.
5878+000090=500208333, &a
6J. If £7i)00 were to be divided between 5 persons.
how nmch ought each person to receive > A71S. it; 1 500.
70. Divide 7560 acres of land between 15 persons.
_,,,.., ^ , ^^?^. Each will have 504 acres.
71. Dmde £2880 between 60 persona.
„r. rrr^ . , ^«' ^acU will reccivG £48.
72. >Vhat 13 the ninth of £972 ? Am. £108.
73. AVhat is each man's part if £972 be divided
among 108 men ? Ans. £9.
74. Divide a legacy of £8520 between 294 persons.
_. ^. „ ^ns. Each will have £29.
7o. Divide 340480 ounces of bread between 1792
persons Ans. Each person's share will be 190 ounces.
/6. Ihere are said to be seven bells at Pekin, each
ot which weighs 120,000 pounds ; if thev were melted
up, how many such as great Tom of Lincoln, wcighinff
9894 pounds, or as the great bell of St. Paul's, in
London, weighing 8400 pounds, could be made from
them ? Ans. 84 like great Torn of Lincoln, with 8904
pounds left ; and lOOlilco the great bell of St. Paul's.
77, Mexico produced from the year 1790 to 1830 a
I>IVIS10N.
09
ive £48.
divided
persons.
Lvc £29.
m 1792
ounces.
:in, eacli
! melted
vcighiug
lul's, in
ie from
th 8904
Paul's.
1830 a
f
I" 5«7,01y,7IO iuil,.3 i, ,' ",t ', ' ""P"™ "l"'"'
ifl29'0775 in 18'57 28b 13o6 m 1740; and
I.alf of ,„en ?i I^ow , ?'" '"'• T ^ '"''"'"■'^ ■""• »
* eo^:"ro:'L^;,t!:rf„,S:!i ^^ ^^'^ '^
I'XAJii.LK.Divldo 84380848 by 87532
87532)84380848(964
5G0204
350128
' the dividend to
vnlaindor after s,h..u.ttitl,o""'''.'l •*''^, *'^^^ ^«) ' 2 (tiio
'"^ f'.'Hried from 98 an.] 1 fT^^" ^? *!,'''°'"' '^ + '^''e 2 to
rowed ulu.n „/ ' ^1 \ *" «<»"ir)en,siito for wl^.A ,„„ i....
lowta When wo coiwideved in ti
e duidund as 10) ; (ti:
19
100
DIVISION.
romalnder when we subtract tlie risht liand digit of 48 from
11 om the 48 )j (tho' remaindcu after eubtractiii;]; the riohf
hand digit ot 67 from 3, or rather 13 iu the dividend), "id
{J times a + the G to bo carried from the 07 + thel for
what wo borrowed to make 3 in the dividend become 13) :
dendT ""^"^^ '''' ^^^^'^ subtracting 79 fro^n 84 in the divi'
j.n;t!,nf'f .^f *' '° .*¥ ^^^^".theses are merely explanatory,
and not to be repeated, the whole process would be
Jmstpart, 4, 18 ; o. 28 ; 2. 48 ; 07 ; G. 79 • 5
Second part, 8. 12; 2. 19 ; 1 32; 0. 45:5 533
lhirdparfc,8;0. 12; 0. 21 ; 0. iO;0. ^5 ; '
J he remainders m this case boincr cyphers, are omitted.
All this will be very easy to the pupil who has prac
tiscd what has been recommended [13, 23, and 651.
I no chief exercise of the memory will consist in recol
lecting to add to the products of the diifcrent parts of the
divisor by the digit of the quotient under consideration,
what IS to be carried from the preceding product, and
I mty bes.deswhen the preceding digit of the dividend
hnnd d?>"''r!u'^ ^^ ^2 5 then to subtract the right
hand di^it of this sum from the proper digit of the
dividend (increased by 10 if necessary)
QUESTIONS FOR THE PUPIL.
1 . What is division ? [66]
3. What is the sign of division ? [68]
4 How are quantities under the vinculum, or united
^^ TAr'°° ""^ multiplication, divided ? [69]
12 nn, r* r *V '?^? ""^'^ *^" ^^'"'' do«« not exceed
12 nor the dividend 12 times the divisor.? [70]
piit Xn .f ' r'^ ^^^. ^^'' ''''''''' ^f '^' different
paits, when the divisor does not exceed 12, but the
divi.lend IS more than 12 times the divisor ? [72 and 771
7. How is division proved ? [79 and 95]. •'
b V\ liat are the rules wlion the dividend, divisor, or
both contain cyphers or decimals.? [80].
9. AV^hat is tho rulo •ii)<l wl^ ^i. ^k. c •.
,.^ "; ^""7 •">5 wJiac are the reasons of 13
different parts, when the divisor exceeds 12 .? [89 and 93j .
GREATEST COMMON MEASURE.
101
10. What is to be done with the remainder > r72
and 89j. ' '''*
11. How is division proved by casting out the nines ?
[95].
12. How may division be abbreviated, when there are
^lecimals i' [96] . '
33. How is division performed, when the divisor is
a composite number ? [97] .
• v^; v!T ,'^ *^^ division performed, when the divisor
i.s but little less than a number which may be expressed
by unity and cyphers ? [98] .
15. Exemplify a very brief mode of performin<r divi
sion. [99]. / °
THE GREATEST COMMOxY MEASURE OF NUMBERS
100.^ To find the greatest common measure of two
quantities —
IluLE.— Divide the larger by the smaller ; then
the divisor by the remainder ; next the preceding
divisor by the new remainder :continuo this process
until nothing remams, and the last divisor will be the
greatest common measure. If this be unity, the ffiven
numbers arQ prime to eack other.
and 4248''' ~^'"*^ *^® greatest common measure of 3252
'"va ?)4248(1
3252
99G)3252('3
2988
264)990(3
792
204)264(1
204
60)204(3
180
"24)6012
48
12)24(2
24
V im
102
GREATEST COMMOxV MEASURE.
fh^i'.*^"'^^''^* remainder, becomes the second divisor 2G4
the second remamder, becomes the third diW&o 19
the last d.v.sor, is the required greatest cLnrS^easuro ^
tied that ' f any quantUv m^ZZ^ ^'' A^' ^^^^^ *° ^^^ «a««
any multiple of t^,2t Xr » thus ff T^n'': %f^} ^^^^'^^^
sum. for if 6 g. into 24, 4 timet and^iS^ ? tL^U ;n/evt
dently go into 24+36. 4+6 times :that is, if ^=^, ^nd^^
6,6+?=4+6. • 6 6
ence between the numbers of ti,^P«u V ^*^^ ""^ *^« ^i^er
due to this difference, l^stte^TtalLr!^ ft^^e ^^
of tmies ..that is, 8ince^i=6, and ^=4, ?^_24 X 36441
oth^rliSSCr ^'^^^^^ e^TeSIHorrect with any
and that it is L ^2^ folo'n 1^^ ^ "^^"^^ ^
we fiXraH 2leTsrer24'^tf"^ ^* *^^ ^"^ «^ t^^« Process,
a multiple of 24?^ thl'slTiSw?' ^'' ^f^^^^ ^* '^
each of them) or 60 and IftThlv 4^^''^''^® '* measures
and 180+24 Twe hnv^ S^ • . because it is a multiple of 60 •
these) or ib4 an Jo^teo'or 2rT^ that it measures each of
pie of 264; and ?92+2ot or 996 ^'hS^QkI^' ^'''''.'' ^ ^^^l^"
and 2988+264 or 3252 Innp J vl' ■ ^^^^' * multiple of 996 :
996 or 4218 (throJlSi v'erLmL?)''^^.^,^^^)!"^ '''^+
each of the given numbls an<f^ fS ^^^^^^'^^^^ it measures
some other be greater ir/n ?tl ^'.^'"^on measure. If not, let
process) measu^r7nf4248 and s'S'TtTS 'T."' *^^ ''^ ^' '^'
measures their difference 996 aniio««' k^' supposition), it
of 996; and. because irmefur'es 3252 n'^^S*^"^ "^^^*^Pl<^
their difference, 284 • an.l 7Q9 tL ' '^'''^ ,^^^^» ^* measures
the difference between j'6 and Tgr^'^of^^^^^^^^^ ^'^^
between 26 4 and 204 or GO • and 1 «0 h ' ""'^ *''^ difference .
ami the difference between'''04 and 1Ro"''"h *" '"^^*'P^« ^^ ««;
a multiple of 24; and tre'd^ffl^l ^'^t^^A^ '^^^^^^^
But measunng 12. it cannot be greatc^, tha^ 12 """ "° "^ ^'
GREATIuST COJIMOxV MEASURE. 103
rntJre\TZlZnZZi'f'T.' *^* '^"^ other common
sequently that 12 s tH 'IS' n ^''' *^''^ ^^~'"^^ ''«"
rule might be inovea fiom r!nv !> '"°" measure. As the
it is true in all Lses ^ °*^'''' ^'^^"^JP^^ ^^"^"y well.
^ 104. We may here remark, that the measure of two
, my quantity, tiio diitit of whoso bwest donnmin„n„„
.3 au even .mn.bor i« aivimhh by 2 at ka t '""
Any number on, ,ng in 5 is divisiblo by 5 at least
j^ Any nttmber onding i„ a cypher is div^isib?^ l'; io at
EXERCISES.
ancllSSV"'lt.f ''' ''"^"^"'^ "^^^'^"^^ •^f 464320
2. Of 638296 and 33SS8 ? ^,w q
3. Of 18996 and 29932? ^t 4
4. Of 2G0424 and 54423 > Ans 9
5. Of 143168 and 2064888 .P Ans'. 8.
t). Of 1141874 and 19823208 > A^Lf. 2.
ihlni^'Jl'Jl ^'''''''' — «« sure of more
of lh4 1 If ''^'"''"''^ "^"'^•''^^•^ a»d a third ; next
ot this hist common measure and a fourth &e S
last common measure found will v.. +i "'^"' ^'^^ J^ie
measure of all the given ^il^^^^ '^' ^''''''' ''^^'^
SOotTnd 673^^^'''^ '^'' ^''^'''^ «^°^^«° «^oasure of 679,
.neatost common measure of 7and 6731 71?' *''^' '^' ■'''"
numhor), for 6734 ^7060 wIH.^? • ^^'"^ remaining
7 is the required numbTr ' ""^ ^•'^•"'•^•igli'. ThereforS
TSoi'^mfHil '"^ "'^ '''' ^^^^^^t ««^'"non ;^a«ure of 0:^0,
104
LEAST C0iv/1\I0N JMULTIPLE.
The greatest common measure of 93G and 73G is 8 and
tlic eonnnon measure of 8 and 142 is 2; therefore 2 is 2o
groutesfc common measure of the given numberr
fncf n toCfa'ft'^ .go through all of themin succession ;
nro to l..S,r„T*i * '* '^ ^^^ greatest common measure, we
used toind tU .T'""'"''''^'"""* '^ ^^^ fi^'«t process, or tlm?
usta to hnd the common measure of the two first numbers
tond proceed successively through all. numDers,
EXERCISES.
i'tLo^^^^jH^^ greatest common measure of 29472
176832, and 1074. Arts. 6. »^'^,
8. Of 648485, 10810, 3672835, and 473580. Am 5
9. Of 16264, 14816, 8600, 75288, and 8472. Ts 8.
THE LEAST COMMON MULTIPLE CF NUMBERS.
titils— ^"^ ^""^ ^^"^ ^'''''* '°'''"'°'' """"^^^P^^ °^ ^^'^ ^^^"
lluLE.— Divide their product by their greatest com
mon measure. Or ; divide one of them by their greate t
common measure, and multiply the quotient by the
o lierthe result of either method will be the required
least common multiple. " 4""^"
KvAMPLE.Find the least common multiple of 72 and 84,
1. is^their greatest common measure.
1^ = 6, and G X 84 = 504, the number sought.
108 Reason of the Rui.E.It is evident that if we muU
SiniT'^i "^""^'r^ *°Sether, their product wHl be a
tiultiple of each by the other [30]. It will bo easv to find
he smallest part of this product, which will stiKe their
of^r'nniilMB ''71 r^"""'^^' '''" 1^^'^^' ^'^^^^ of t'^e factors
Irotlot^nf^Xr f ^ ''"•X ""'"^^^' '^"'i multiplied by the
& J V 1 ,""'■, ^'"'^'''■'' '' "^'""^ to the product of all the
.io.« ^nidcl by the same number, ironco "'>. .pd ri L; !
LEAST COMMON MULTIPLE.
106
$m
2X84
~^^ (the nineteenth part of their producc)=I?x84, or 72 x
J. y
u
_. Now if ' and __ be equivalent to integers, ^x84 will be a
multiple of 84, and°x72, will be a multiple of 72 [29] ;
andig_, L.X84, and 72 X~ will each be the common
multiple of 72 and 84 [30]. But unless 19 is a common measure
of 72 and 84, j^ and _ cannot be both equivalent to integers.
Tl.erefoie the quantity by which we divide the product of the
gixen numbers, or one of them, before we multiply it by the
ctlior to obtain a new, and less multiple of them must be he
c.mmon measure of both. And tbe multiple we obtain w 11
cvi.lent y, be the least, when the diviso ■ we select ^stlo
grea es quantity we can use for tiie purposethat is! e
greatest common measure of the given numbers
It follows, that the least common multiple of two
numbers, prmie to each other, is their product.
EXERCISES,
1. Find the least common multiple of 7S and 93
Ans. 2418. ^
2. Of 19 and 72. Ana. 1368.
3. Of 464320 and 18945. Ans. 1759308480
4. Of 638296 and 33888. Ans. 2703821856."
5. Of 18996 and 29932. Ans. 142147068.
6. Of 260424 and 54423. yl%5. 1574783928.
109. To find the least common multiple of three or
more numbers —
KuLE.— Find the least common multiple of two of
thcin ; then of this common multiple, and a third ; next
ot this last common multiple and a fourth, &o The
last common multiple found, will be the least common
multiple sought.
TAAMPLE.Find the least common multiple of 9, 3, and 27
^ '^ IS the greatest common measure of 9 and 3 ; therefore
g X 3, or 9 is the least common multiple of 9 and 3.
'^^9 is the greatest common measure of 9 a«7 ; therefore
^ X 9, or 27 is the required least common multiple.
I,
ii J; iii
106
LEAST COMMON MULTIPLE.
that^^ili'lir/"'.'""*^ RuLE.By the last rule it is evident
tnat J7 IS tl e least common multiple of 9 and 27. But since
a nuauri* %"•' 'IV'V, ^^'"/'^ .^^ multiple of U, mu^t^airt:
tlii is Imanif ."'"'i" ^'"''' ""'""^"^ "^"^*'P^«' ^e^a'^so none
tnat IS smaller can be common, also, to both 9 and 27 since
they were found to have 27 us their least common multiple
EXERCISES.
it
aIs. mls'^'^ ^'''''^ ''''"'""'''' multiple of 18, 17, and 43.
n Ri .^?' 3 ^^^ ^"^ ^1 '^>^' 1265628.
10 ^;^^«?f ?'.rp,f ^' ^^^ ^^^2. .in.. 2937002688.
10. Of 53/842, 1G81<J, 4367, and 2473.
11 Of oir'jp o.iot. n... ^«5. 8881156168989038.
li Of 21636, 241816, 8669, 97528, and 1847
Ans. 1528835550537452616.
' QUESTIONS
1. IIow is tho greatest common measure of two quan
tities found .? [louj. ^
2. What pvinciples are necessary to prove the correct
ness ot the rule ; and how is it proved ? [101, &c 1
o. llow IS the greatest common measuie of thiee, or
more quantities found .? [105]. '
4. How is the rule proved to be correct ? [1061
o liow do we find the least common multiple of two
numbois that are composite .? [107].
6. Prove the rule to be correct [lOS].
7. How do we find the least common multiple of two
prime numbers .? [108.]
5. How is tho least common multiple of three or
more numbers found.? [109].
9. Prove the ;ule to be correct [110].
^ In future it will be taken for granted that the puB^
IS to be asked the reasons for each rule, &c.
t
107
SECTION III.
19 1
m
quan
REDUCTION AND THE COMPOUND RULES.
The pupil should now be made familiar with most of
the tables given at the commencement of this treatise.
"^ REDUCTION.
t
1. Reduction enables us to change quantities from
one '^nomination to another without altering their
value. Taken in its more extended sense, we have often
nractised it already : — thus we have changed units into
tens, and tens into units, &c. ; but, considered as u
separate rule, it is restricted to applicate numbers, and
is not C0nfin4 to a change from one denomination to
the Tiexi higjier, or lower
2. Reduction i» either descending, or ascending. It
is reduction descending when the quantities are changed
from a higher to a lower denomination ; and reduction
ascending when from a lower to a higher.
JRjcduction Descending,
3. RuLE.^^Multiply the highest given denomination
by that quantity which expresses the number of the
next lower contained in one of its units ; and add to
the product that" number of the next lower denomina
tion which is found in the quantity to be reduced.
Proceed in the same way with the result ; and continue
tlie process until the required denomination is obtained.
Example. — Reduce £6 16s. OjcZ. to farthings.
£> s. d.
6 „ 16 „ Q\
136 shilling8=£6 „ 16.
12
1632 pence = £6 „ 16 „ 0.
4
C520 farthings OB jC6 „ 16 „ Q\.
" * 1
l a— l: :^'i
" 11 ■!
108
REDUCTION.
6 aro 24, and 1 nrA 9*^ 4 .. ^"^ ^ ^^°?6« ^ ^ o 12. 4 tunes
4 SrS m'ny' fertile ."^"^"^'""'^.■'™™• pence to
EXERCISES.
93312^'''" '^''''^ ^^'^^""°' ^*" ^^^~S P^^^e^ ^^«.
2. How many shillings in i2348 > Ans. 6960
^. How many pence in ^638 10^. ? Ans. 9240
4. How many pence in ^58 13.. ? Ans. 14076
5. How many farthings in £58 135. ? Ans. 56304
67291 "'"""^ farthings in £59 13.. G^d. ? Am,
7. How many pence in £63 0.. 9d. ? Ans. 15129.
a!s\ 1864. ""'"^ P'""^^ ^" '' ^^*> ^ ^^^^ 16 ife ^
.l.^: 1^68. '"''''^ ^'"''^' ^"^ ^'^ '^*' ^ "i"' 1^ J'^ ^
^^^5^, ?°^ "^^"y grains m 3 lb., 5 oz., 12 dwt 16
IlKDUCTION.
109
11. How many grains in 7 lb., U oz., irrdwfc./M
grains ? Aiis. 45974.
12. How many hours in 20 (common) years? Am
175200.
13. How many feet in 1 English mile ? Ans. 5280
14. How many feet in 1 Irish mile > A7is. 6720
15. How many gallons in 65 tuns .? Ans. 16380
16. How many minutes in 46 years, 21 days, 8 hours,
56 mmutos (not takmg leap years into account) ? Ans.
*4208376.
17. How many square yards in 74 square English
perches ? Ans. 22385 (2238 and one half).
IS. How many square inches in 97 square L'ish perch
es.? Ans. 6} 59H88. ^
19. How many square yards in 46 English acres, 3
roods, 12 perches ? Ans. 226633.
20. How many square acres in 767 square English
miles ? Ans. 490880. ^
21 . How many cubic inches in 767 cubic feet ? Ans.
I32o376.
22. How many quarts in 767 pecks .? Ans. 6136
23. How many pottles in 797 pecks ? A71S. 3188.
Reduction Ascendviw.
5.
required denomination is
. hings to pounda, &c.
EuLE.— Divide the given quantity by that number
ot its units which is required to make one of the next
Higher denomination— the remainder, if any, will be of
the denomination to bo reduced. Proceed in tlie same
manner until thi ' ' 1 o ■ . , ,
obtained.
Example.— Reduce 8(.,,
4)856347
12)214086f
892 ; „ 01=856347 farthings.
4 divided into 85G347 farthings, gives 214086 ponce arul
3 farthings 12 divided into 2!4ol6 pence, gives 17840
'i«.lo°^' r^ ^ ir^V,9« 20 divided into 1^840 shillings, a<ives
7}7xr ''" i""" «^iiJ^ings; there ie, therefore, nothing in the
Bhilhngs' plac . of the result. **
no
HKDUCTION.
Wy (livi.r, by 20 if ^0 (llvi.lo l.y 10 lui.l 2 FSoc If 971
any, [N,o. 34] w uch will then bo tbe unit« of sbilli, .^s
11 t b., result ; and tie quotient will bo tens of shillings :^
i.viding tbe xa tor by 2 gives the pounds as quotion? and
di;;:srlS^'^' '' ''^^^ ^^ ^  ^^ Quired q.«..
VoIIk^"''"'"'" "*^ T"^ Rin.K.It is evident that every 4
Earthings are equivalent to one penny, and every 12 pence to
«neslnll.ng,&c ; and that ^hat is left after taking away 4
far b.nga as often as possible from the farthings? intTsfbu
farthings, what remains after taking away 12 pence as often
as possible from the pence, must be pence, &o ^
7. To prove i?€r/%c^207i.— Reduction asccndin'' and
acscendiDg prove each otljer. ^
Reduction <
f
* s.
20
417
12
d.
farthirgs.
4)20025
Reduction j 12)5006
20)417'„ 2
5006
4
4)20025
Proof .
Proofs
1
12 )5006
2 0)417 „ 2
^20 „ 17 „ 2]
^20 „ 17„ 2«
20
417
12
20025 farthings.
RXERCISES.
nooto ^^"""^ "'''''^ P^'^^^ ^" ^3312 farthings ? Ans.
25. How many pounds in 6960 shillings? Ans ^2348
roon a7 ""^""^ P''''''^'' ^'^^ ^^ ^^^ halfpence ? Ans.
pi/fi Us, od.
JI\ ^""Z^^"^^ ^^^"^^^^ &c. in 7675 halfpence ? Ans.
Jblo 19*. 9^a.
28. How many ounces, and nounds in 4352 drams?
Alls, 2/2 02., or iv ib.
REDUCTION.
Ill
29 How many cwt., qis., and pouuda in 1864 pounds >
Ans. l(i cwt.,2(ir,s., 1(5 lb. i ".
30. How many hundreds, &c., iu 16G8 pound.s. A,ls
14 cwt., 3 qns., i6 lb. ^
31. How many pounds Troy iu 115200 frvahiH >
Ans. 20. " b •
J ^^vIm'^^ '""""^ 1'°''°'^'* '" ^^'^''^20 ^^' avoirdupoise ?
tills. d720.
.,.2^1* ^?^^ J"*"^ lio<rsheads in 20C58 gallons ? Ans.
127 hogsheads, 57 gallons.
34. How many days in 87G0 hours ? Ans. 365
35. How many Irish miles iu 1S34560 feet.? Ans.
i lO.
^^ ^low many English miles in 17297280 inches ?
37 How many English miles, &c. in 4147 yards >
Ans. 2 miles, 2 furlongs, 34 perches.
iiS. How many Irish miles, &c. in 4247 yards ? Ans
i mile, 7 furlongs, 6 perches, 5 yards.
39. How many English ells iu 576 nails ? Ans 28
«ils, 4 qrs.
40. How many English acres, &c. in 6097 square
yards ^ A71S. 1 acre, 8 perches, 15 yards.
41 How many Irish acres, &c. in 5097 square yards ?
Ans. 2 roods, 24 perches, 1 yard.
. 42. How many cubic feet, &c., in 1674674 cubio
inches ? Ans. 969 feet, 242 inches.
43. How many yards iu 767 Flemish ells ? Ans
07o yards, 1 quarter.
44. How many French ells in 576 English ? Ans. 480
.f i/\ T^J^^ J'^'^^' *^^" '''^^ ^^'"^^^^ of a pound
of gold, to farthmgs ? Ans. 44856 farthin^^s
46 The force of a man has been estimated as equal
to what, in turning a winch, would raise 256 lb in
?'"''?nl'/^'^ ^^' ^" ^'"^Sing a bell, 572 lb, and in row
mg, 608 lb, 3281 foet in a day. Uow man>^ hundreds
quarters, &c., in the sum of all these quantities ? An^'
16 cwt., 2 qrs., 7 lb.
47. How many linos in the sum of 900 foet, tha
i
112
nzDvvvioti.
length of ♦ho tomple of tho sun at Balboc, 450 foot its
breadth, 22 foot tho circuruforuiiee, and 72 feet tho
height of many of it« columns ? ylns. 207936
, 48. How many square toot in 7tJ0 English acres, tho
inclosuro m which tho porcelain pagoda, at NanKiiiir,
4J. rho great boll of Moscow, now lying in a rit
qfionnnT /"'•' supported it having boon burned, weighs
36)000 lb. (some say much more) ; how many tons, &o ,
m this quantity ? A7i^. 160 tons, 14 cwt., 1 qr., 4 lb.
QUESTIONS FOR THE PUPIL.
1. What is reduction .' [1].
^ 2. What is the difference between reduction descend
ing and reduction ascending > [2] .
3. What is tho rule for induction descending ? fal
4. What IS tho rule for reduction ascending ? fsl
5. How is reduction proved ? [7].
Qiiestiom fonmdcd on the Talk page 3, c^.
6. How are pounds reduced to farthings, and farthincrs
to pounds, &c. ? ^ ' t3
7. How are tons reduced to drams, and drams tc
tons, &c. .''
8. How arc. Troy pounds reduced to grains, and
grams to Troy pounds, &c. } b j ^
9 How aro pounds reduced to grains (apothecaries
weight), and grams to pounds, &c. ?
10 How are Flemish, English, or French ells, re
eHs'' &o'' T ' ""'* '"''''''''' ^'^ i^lemish, English, or French
^^11. How are yards reduced to ells, or ells to yards,
12. How arc Irish or English miles reduced to linos,
or lines to Irish or English miles, &c. > '
13. How are Irish or English square miles reduced
to square mehes, or square inches to Irish or English
square miles, &c. ? ^
n
450 foot its
72 feot tho
m.
1 acres, tho
NanKing,
35000.
[? in a j)It
nod, weighs
y tons, &o.,
qr., 4 lb.
COMPOUND RULES.
113
a descend
er
? [5].
S^'C.
1 farthings
drains tc
lins, and
itliccarios
ells, rc
>r French
to yards,
to linos,
reduced
English
14. Ilowaro cubic feet reduced to cubic inches or
cu»)io inches to cubio feet, &c ^ "loncs, or
U,,!i;«.'':T "'•" """' ""'"™^' "• """«■'. or naggin, .o
lu,',»; "■"' "™ •'""'' "^""«^ '" 8"''™^ <" gal'ons to
and';:,":: i:JX ^''^ •"■•") J"'' to pint,,
yoa'^i &o!" ""■" ^"™ '"''"""'' '" ""^•'» 0' tlmU, to
or Jw'"t:i';;;!:f:T<,.<f "■» <='™'") ^'o'' '° tWrd.,
THE COMPOUND RULES.
^. The Oonipound Rules, are those which relate to
apphcate nuinbcvs of more than one denomination.' *'
conin 1 i " ^^ '"o.u^y, wcnghts, and measures, wore
coast, ucted according to tlie decimal sv.stcm, on v the
T ?' ^^S^"P^« ^^^^'^'tion, &c., would be' Sir d
to d Z \% ' ««"«i^l«'ble advantage, and ^ a ''
tond to snuplify mercantile transactions If i o f • •
things were one penny, lu pence one shillincr and To
shil in,g. one pcnind the addition, for exan";!^ f i?
a noun i \l \^^' u''^ ^V'''' ^^^^^^^^ to eparate
Id OikU r ^'" "?^ of comparison," from its parts!
bots'follow^^^^^"^ * " ' *^^^^^^ '' ^ P"^)> 'Id
1983
6865
Sum, 8848
The addition might be performed by tho ordinarv
rules, and the sum read off as follows " mV . ^'^'^'"''/y
ci^hfc sliillings, four pence, ilnd ^ 7.,^ ^^ K^'j; ^
even with the present arrangemoiU of mo ; wei. i
tt^ir^ the rules alr^uly given for ad^Ji""^^'
ttr un/^ ' ""'f"^ "'"^^^ liave.b.en made to include
he a d.tion subtraction, &c., of .r.nlicate .^ZZ
can...tmg of more tlian one denomination ; sin"cc"'thQ
l!
¥■
p
w
114
COMPOUND ADDITION.
principles of both simple and coiDpound rules are pre
cisely the sa.nothe only thing necessary T bcnr
carefully m n.md, being the number of any mo de!
nomination necessary to constitute a unit o/tl next
COMPOUND ADDITIOX.
i^nl' nfl'^^'~^' ^i''^ ^'''7'' ^^^« ^'^^^euds so that qnantU
vertical oIuZ ''T'T'''' "^'^^^ «^^^"'^ ^'^ ^^« «
wi nf ^^^•^"^nunits of pence, for instance, under
units of pence tens of pence under tens of pence un ts
01 shilWs under units of shillings, &c ^ '
U. Draw a separating line under the addends.
aenoLi^it:^,:^^!^^^^ the same
pence, ^., begSnin^wilh IhrLtst^^^^'""^^' ^^"^^ '^
ber of tl .rl? '^"\^^^;;«J c«'""m be less tlian tlie num
toei ot that denomination which makes one of the m vf
^ s. d.
52 17 33 )
6} } addends.
47
60
5
14
2' \
IGG 17
! and I make 3 farthings, which w^Mi ^ m..i.« r r
a 2 are 3, and « arc 9, aad 3 are 12"pro/i:;,'uaf tJ'S
Ill OS are pre
!<'iry to bear
any one do
of the next
that qnanti
iu tho same
mice, imJer
pouco, units
mds.
f the same
;«, pouco to
n tlic nura
)f tho next
ot, for each
nomination
one to tho
under the
nomination
it in the
47 5s. G}/1.,
ike () far
denoiuiiiii
10 jiresont.
uarricii)
ual to one
COMPOUND ADDITION. Hq
of the next denomination or thof nf c1,it . ,
and no pence to be sriow 1 t^r""'' ^''^ ^^"'i'^d,
in the pence' place of the su n 1 sti r ""''z?",^ '*' ^^^P^*^^
and 14 are 15,^and 5 are 20 ^d 17 n i ^ iu^^ ''^'''^^)
to one of tl. next ck^nomLtl or hat'of n "?.'~r^r
earned, and 17 of the present or thnfViFn""'*'' *^ ^^
set down. 1 pound anf 6 l^eS^'andl are if 3' 9^ ^'
10 pounds— equal to unit, nf Z^ j ^ . • "'"' 2 are
1 t'u of pounds be cm ed l"^?™ ' w''" "'',''''''"■ »d
11 and 5 L 16 tens ofTonSd^ VZ^tZT ' ""' " ""
thoVeaS;t;or[st r;Xt It' TT
not so necpssflrv fn %.„+ T i' ■•■* ^s evidently
Of all the sunis. "^ atterwards find tne amount
Example :
£
s.
d.
57
14
21
32
10
4
19
17
6
8
14
2
32
5
9j
47
6
4)
32
17
2
5(J
3
9
27
4
2r
52
4
4
37
8
2
= 151 7 11
^ s. d.
404 11 10.
= 253 3 11
a dlt .?'nfr '''^'^'''^ "'"'^ ^°^"^"' ^« "laj put down
«rpl?l' ""'""" ""^ ' o™ dot "U^ing ti; S
I
I
116
COMPOUND RULES.
&
s.
(/.
67
•14
2
32
10
4
19
•17
•G
8
•14
2
32
5
•9
47
•6
4
32
17
2
56
•3
•9
27
4
2
52
4
4
37
8
2
404
11
10
2 pence and 4 are 6, and 2 are 8, and 9 are 17 pence
equal to 1 shilling and 5 pence ; we put down a dot and carry
?• , ^,«:"4 2 are 7, and 4 are 11, and 9 are 20 penceequal
to 1 shilling and 8 pence; we put down a dot and carry 8.
« and 2 are 10 and are 16 penceequal to 1 shillino; and
4 pence ; we put down a dot and carry 4. 4 and 4 are 8 and
'T"" 1^— which, being less than 1 shilling, wo set down
under the column of pence, to which it belongs, &c. We find
on addmg them up, that there are three dots: Ave therefore
carry o to the column of shillings. 3 shillings and 8 are 11,
and 4 are If, and 4 are 19, and 3 are 22 shillingsequal to
1 pound and 2 shilhngs; we put down a dot and carry 1.
1 and 1< are 18, &c. ^
Care is necessary, lest the dots, not being distinctly marked,
may be considered as either too few, or too many. Thia
method, though now but little used, seems a convenient one.
14. Or, lastly, set down the sums of the farthinfrs,
shillmgs,_&c., under their respective columns; divide
the tarthmgs by 4, put the quotient under the sum of tlio
pence, and the remainder, if any, in a place set apart
tor It m the sum— under the columu of farthings : add
together the quotient obtained from the farthiSgs and
the sum of the pence, and placing the amount under
the pence, divide it by 12 ; put the quotient under the
sum of the shilbngs, and the remainder, if any, in a
place allotted to it in the sura— under the column of
pence ; add the last quotient and the sum of the shil
lings, and putting under them their sum, divide tho
latter by 20, set down tho quotient under the sum of
COMPOUND ADDITION.
117
17 pence—
•t and ciury
nee — equa!
id carry 8.
hilling and
I are 8 and
) set down
. We find,
e therefore
d 8 are 11,
— equal to
id carry 1.
ly marked,
ny. This
snient one.
farthinfi:s,
i ; divide
um of tlio
set apart
ngs; add
lings and
nt under
mder the
my, in a
)lumn of
the shil
vide tho
sum of
■
the pounds, and put tho remainder, if any, in the sum—
under the column of shillings ; add the last quotient
and tho sum of the pounds, and put the result under
the pounds. Using the following example —
£> s. d.
47 9 21
362 4 in
51 16 2
97 4 G
541 13 2i
475 6 4
6 11 11.1
72 19 9,^
1G51 82 47 13 farthings.
4 4 3
86 50
1055 G 2!
The sum of the forthingg is 13, which, divided by 4, give.i
3 its quotient (to be put dowia under the pence), and ouh
farthing as remainder (to be put in the sum total— under
the farthings). 3,'/. (the quotient from the forthings) and
47 (the smn of tlie ponoc) are 50 pence, which, being put
down and divided by 12, gives 4 shillings (to be set down
under the shillings), and 2 pence (to be set down in tliu
sum total— under the ponce). 4.^. (the quotient from the
ponce) and 82 (the sum of the shillings) are 86 shillings,
which, being sot down and divided by 20, gives 4 pounds
(to be set^down inider the pounds), and G shillings (to bo
sot down in the sum total— under tho shillings). £4 (tho
quotient from tho shillings) and IfuU (the sum of tho
pounds) aro 1G55 pounds (to be set down iu the sum total
under the pounds). The sum of the advlendss ?s, therefore,
found to be j£lG55 6s. 2(/.
15. In proving the compound rulo«(, wo can geuorally
avail ourselves of tho methods used with the sin.i^l<? vul,^
[Sec. IT. 10, &c.]
11
m
ssM
(18
COMPOUND ADDITJON.
£ 8. d.
70 4 6
57 9 9
49 10 8
183 4 11
£ s. d
674 14 7
466 17 8
676 19 8
627 4 2
KXERCISES FOR THE PUPII,
Money.
d. £ s. d.
7 76 14 7
6 67 16 9
8 76 19 10
£
58 14
69 16
72 14
* s. d.
767 15
472 14
567 16
6
6
7
423 3 10
£ s.
567 14
476 16
647 17
527 14
d.
7
6
6
3
(4)
£
». d.
84
8. 2
96
4 Oi
41
6
(8)
£
.V. d.
327
8 6
601
2 111
864
6
121
9 84
£ s. d.
4567 14 6
776 16 7
76 17 9
51 10
44 5 6
(10)
£ s. a
76 14 7
667 13 6
67 16 7
6 4
5 3
2
4
^ s. d.
3767 13 11
4678 14 10
767 12
10 11
8 4
9
6
11
(12)
£ s.
6674 17
4767
3466
6r^4
8762
d
16 Hi
17 101
2 24
9 9
£ s. d. £ s d
9767 6i 6767 11 ei
7649 11 2i 7676 16 94
4767 16 101 5948 17 sl
164 1 1 6786 7 6
92 7 24 6326 8 24
(15)
£ s. d.
6764 17 6
7457 16 5
6743 18 04
67 6 6k
432 6 9
« (16)
£ s. d,
634 7 114
65 7 7
7 12 lOi
5678 18 8
439
« (17)
* s. d.
14 71
677 1
6767 2 6
8697 14 74
6684 0
(18)
* s. d.
5674 16 7i
4767 17 61
1645 19 7i
3246 17 6
4766 10 5
(19)
£ 8. d.
6674 1 94
4767 11 10^
78 18 Hi
19 104
6044 4 1
£
4767 14
743 13
7674 14
7 13
760 6
(20)
*. d.
7i
74
6i
84
4
5(
6i
34
COMPOUND ADDITION,
119
(21)
£> s. d.
674 11 11.1
667 14 10
476 4 11
347 15 Oi
476 18 94
(25)
£ s. d.
576 4
7 7
732 19 04
667 9^
764 2 64
n
6
(22)
£> s. d
476 14
576 15
76 17
576 11
463 14
7
H
n
8
94
(26)
* s. d.
549 4 6i
7 19 91
16 64
734 19 9i
666 14 44
(23)
£ s.
d.
674 13
Bk
45 16
74
476 4
61
577 16
04
678 6 8
(27)
£ s. d.
876 3
5
66 11 11
123 6 24
12
(241
£
s.
d.
674 17
6^
123
12
2
667
7*
679 18
91
476
6
64
(28)
— ••
£
8.
d
219"
6
32 11
8^
04
127
8
2
29
6
6i
(29)
Jlvoirdupoiae Weight.
(30)
(31)
"nT ^l T/¥ ,5 ^^^^l .^ cwt.qS «>
37
14
2
1
14 44
15 66
11 47
1
3
1
16
11
16
34 3 17 66
37 1 16 57
47 2 27
3 14
1 17
58 2 26
128
12
(33)
(34) (35) (86)
tT^^.t'^iur.r^.^j.^v
66
3
r i»
13
69
2
17
20
476
764
3
1
47
2
17
3
6
o
81
2
14
67
1
15
1
15 667 2 19
7 4 1 20
14 67 3 2
18 767 1 n
7i
74
6i
34
4
777' T ,■? i?.'F%i? .T'^^ J> .^S
767
44
567
676
341
1
1
3
1
2
16 476 "1 24i 447
J 7 766 3 214 676
13 767 1 16 467
667 2 15 563
11 973 1 12 428
1
1
1
1
7
6
7i
6
04
14 12
8 4
7
5
fl>
12
7
15
8
14
IS t,r
130
COMPOUND ADDITION.
lb
7
6
9
ib
67
07
66
74
12
(41)
Troy IVcight.
(42)
°n" *^7* ^ff • ? °^ d^t grs. lb
U 6 9 6 9  o
6 6
6
7
8 8
7
7
6
G
7
4
88
80
(43;
OB. dwt. gra.
7 9 ^8
9 8
8 7 6
21 11 18
9
9
8
6
3
(44)
dwt
12
11
10
6
6
(45)
14 87
11
5
3 44
4 07
oz.
dwt.
gfs.
lb
\J
7
12
67
11
12
3
16
14
40
12
10
13
22
8
9
10
11
(40)
oz. dwt. grs.
10 14 ^1
11 :.
9
7 e
18 14
9
8
10
(47)
yds. qrs. nls.
99 3 1
47 1 3
70 3 2
Cloth Measure.
(48)
yds. qrs. nls.
176 3 3
47 2
7 3 3
(49)
(50)
y^' *^o^ "S^' y^« <!"• nls
37 3 2 2 1
2 3
2
224
6 3 2
3
(63)
i,51) (52)
UTlTTSiiifr
54 3 673 2 3
ts.
*»9
80
98
87
41
407
1
173
148
92
1
2 1
3 2
(55)
hhds.
gls.
ts.
3
9
89
39
7
3
40
70
2
27
44
1
20
64
TVine Measure.
(56)
hhds. gls.
3 3
3 4
1 56
2 7
2 17
(57)
ts. hhds. gls.
76 3 4
67 3 44
1 66
5 3 4
G02 27
sn
CX)MPOUND ADDITION.
121
(43;
E. dwt.
grs.
7 9
8
9 8
i 7
6
(46)
. dwt.
grs.
> 14
11
11
^•
9
7
5
18
14
. (50)
as. qrs. nls.
2 1
6 3 2
3
(54)
i. qrs. nla.
6 1 1
6 3 1
110
3 2 3
57)
ihds. gla.
4
44
66
4
27
8
8
1
3
Time.
(58) (59) (60)
^99 st \' Z ff • t K' ^f: y ± ^. .
99 859 9
88 8
77 120 7
66
67
49
265 115 2 42
60
6
90
76 1
3
1 2
60
67
68
69 127 7
120 9
70 121 11
6 47 3
8 9 11
60
44
44
41
17
61. What is the sum of the following :— three hun
dred and ninetysix pounds four shillings and two pence •
five hundred and seventythree pounds and four pence
halfpenny ; twentytwo pounds and three halfpence •
four thousand and five pounds six shillings and three
farthings.? Ans. iE4996 IO5. S^d.
62. A owes to B ^£567 16s. 7Jrf. ; to ^£47 I65 •
and to D ^56 Id. How much does he owe in all ?
Ans. iE671 12s. S^d.
63. A man has owing to him the following sums •
^3 10s. 7d. ■ £46 l\d. ; and ^52 14s. U. How much
IS the entire .? Am. £102 5s. ^\d.
64. A merchant sends ofi" the following quantities of
hutter :— 47 cwt., 2 qrs., 7 ft, ; 38 cwt., 3 qrs., 8 lb ;
and 16 cwt., 2 qrs., 20 lb. How much did he send off
m all .? Ans. 103 cwt., 7lb.
65. A merchant receives the following quantities of
tallow, viz., 13 cwt., 1 qr., 6 ib ; 10 cwt., 3 qrs., 10 ft,:
and 9 cwt., 1 qr., 15 ft,. How much has he received in
all.? ^?is. 33 cwt., 2 qrs., 3 ib.
66. A silversmith has 7 ft,, 8 oz., 16 dwt. ; 9 lb 7
oz., 3 dwt. ; and 4 ib, 1 dwt. What quantity has he >
Ans. 21 ib, 4 oz. ^ j
67. A merchant sells to A 76 yards, 3 quarters, 2
nails ; to B 90 yards, 3 quarters, 3 nails ; and to C, 190
yards, 1 nail. How much has he sold in all .? Ans 357
yards, 3 quarters, 2 nails.
68. A wine merchant receives from his corj^spondent
4 tuns, 2 hogsheads ; 5 tuns, 3 hogsheads ; and 7 tuns,
1 hogshead. How much is the entire > Ans. 17 tuns
2 hogsheads. '
II;
122
COMPOUND ADDITION.
.69. A man has three farms, the first contains 120
vf ?' on '''°^?' '^ P^'^h^^; the second, 150 acres, 3
roods 20 perches ; and the third, 200 acres. How much
land does he possess in aU ? Ans. 471 acres, 1 rood, 27
perches. ' ^.^u, ^/
70. A servant has had three masters ; with the first
He lived 2 years and 9 months; with the second, 7
years and 6 months ; and with the third, 4 years and 3
months. What was tlie servant's age on leaving his
last master, supposing he was 20 years old on going
to the first, and that he went directly from one to the
otHer .? Ans. 34 years and 8 montlis.
nJi' Ft^ "^^°y "^^y^ ^^^"^ *^« 3rd of March to the
23rd of June ? Ans. 112 days.
72. Add together 7 tons, the weight which a piece
ot far 2 inches m diameter is capable of supporting • 3
tons, what a piece of iron onethud of an inch 'in
diameter will bear ; and 1000 Jb, which wiU bo sustained
by a hempen rope of the same size. Ans. 10 tons, 8
cwt., 3 quarters, 20 ib. '
73. Add together the following:— 2^., about the
value of the Roman sestertius ; 7i^., that of the dena
rius; lid., a Greek obolus; 9d., a drachma: £3 15s
A.d4^ 6s. 9rf., the Jewish talent. Ans. £bl\ 2s
74. Add together 2 dwt. 16 grains, the Greek drachma:
1 lb, 1 oz., 10 dwt., the mina ; 67 ib, 7 oz., 5 dwt., the
talent. Ans. 68 ib, 8 oz., 17 dwt., 16 grains.
QUESTIONS FOR THE PUPIL.
1. "What is the difierence between the simple and
compound rules ? [8] .
2. Might the simple rules have been constructed so
as to answer also for applicate numbers of difierent
denominations.? [8].
3. What is the rule for compound addition ? [9].
4. How is compound addition proved > [16].
5. How are we to act when the addends are numer
ous ? [12, &c.]
tlili
COMPOUND SUBTRACTION.
jxins 120
acres, 3
ow much
rood, 27
tlie first
2cond, 7
•s and 3
ving his
)n going
B to tllQ
I to the
a piece
'ting; 3
inch in
istained
tons, 8
»ut the
6 dena
'3 15s.
el ; und
achma;
7tj the
^ 123
ie and
ted so
fferent
umer*
COMPOUND SUPTRACTION. .
16. I^ULEI. Place the digits of the subtrahend
under those of the same denomination in the minuend^
tarthmgs under farthings, units of pence under units of
pence, tens ot pence under tens of pence, &c.
II. Draw a separating line.
* ™ ^"^^^act eacli denomination of the subtrahend
trom that which corresponds to it in the minuend
begmnmg with the lowest.
,, ^7'J{ ^"^ denomination of the minuend is less than
that of the subtrahend, which is to be taken from it,
add to It one of the next higher— considered as an equi
valent number of the denomination to be increased :
and, either suppose unity to be added to the next deno
mmation of the subtraliend, or to be subtracted from
the next of the minuend.
V. If there is a remainder after subtracting anr
denommation of the subtrahend from the corro.?pond
lug one of the minuend, put it under the colmnn which
produced it.
yi. If in any denomination there is no remainder,
put a cypher under it—unless nothing is left from any
higher denomination. ^
17. Example.— Subtract £56 13s. 4^,d., from £96 75. 6c/.
£> s. d.
96 7 6, minuend.
5613 4, subtrahend.
39 14 11, difference.
We cannot take ^ from , but— borrowing one of the
pence, or 4 farthings we add^it to the I and then say 3 far!
things from 5 and 2 farthings, or one halfpenny, remains •
we set down i under the farthings. 4 pence ^om^/w«
have borrowed one of the 6 peSce), anrone pcnny^Je
naitpence ) 13 shillings cannot be taken from 7 but ^hor
nC of /S^ ''T° r ''* ^°^° 14 i^ «^e shillinl'
place of the remainder. 6 pounds cannot be tak^n fmnf ^
K^y^ nave borrowed one of the 6 pounds in the minae^)
$
124
COMPOUND SUBTRACTION.
})nt 6 from 15, and 9 remain: we put 9 under tlio units <.f
one ol the 9), and 3 remam: we put a in tlio leas of pouud.i'
place of the icuiamdiT. ^
«nnf; „T^*\^ """^^i ""*? ^^'^^ ""^"'""'^ ''f '* '^''« substantially the
same as those already given for Siniple Subtraction [Kec. 11
J t, &c. J It 18 evidently not so necessary to put down cvDliera
Where there ,s nothing in a denomination of the mlluff
suiLetrnTscc. U^Ol ''''''' '" ''' ^^"^ "^^ ^ «'P^°
£ «. «?.
From 1098 12 6
Take 434 15 8
663 18 10
KXERC1SK8.
* s. d.
7G7 14 8
486 13 9
£ s. d.
70 15 6
14 5
£ «. d.
47 16 7
39 17 4
£ .?.
97 14
6 15
(>
7
From
& s.
98 14
Take 77 15
^ s. d.
47 14 6
38 19 9
* s. d.
97 10 6
88 17 7
£ s. d.
147 14 4
120 10 8
(10)
£ s. d.
6()0 15
477 17 7
£ s. d.
Prom 99 13 3
Take 47 16 7
« (12)
£ s. d.
*l^l 14 hh
476 74
(13)
* *. rf.
891 14 li
677 15 61
(U)
£ v. <;.
676 13 7^
467 14 92
« (15)
£ s. d.
From 667 11 6^
Take 479 10 10^
(16)
£ s. d.
971 Ok
7
(17)
£ s. d.
437 15
11 14
(18)
£ .V. d,
478 10
47 11 0^
(19)
cwt. qrs. lb
From 200 2 26
Take 99 8 15
100
II
Avoirdupoise Weight.
(20)
(21)
cwt. qrs. lb
cwt. qrs.
lb
275 2 15
9064 2
25
27 2 7
9074
27
(22)
cwt. qrs. fl^
654
476 3 5
f
ft)
from 554
fake 97
COMPOUND SUBTRACTION.
Troy Weight.
(23) (24)
oz. dwt. gr. It) oz. dwt. gr.
9 19 4 946 10
16 15 17 23
125
(24)
lb oz. dwt. gr.
917 14 9
798 18' 17
457
9 2 13
Wine Measure.
(26) (27) (28) • (29)
ts. hhds. gls. ts. hhds. gls. ts. hhds. gls. ts. hhds. gls.
From 81 3 15 64 27 304 64 66 1
Take 29 2 26 3 42 100 3 51 27 2 25
2 52
Time.
(80) (31) (32)
yrs, ds. lis. ms, yrs. da. hs. ms. yrs. ds. hs. ras
From 767 131 6 30 476 14 14 16 567 126 14 12
Take 4T6 110 14 14 160 16 13 17 400 15
291 20 16 16
33. A shopkeeper bought a piece of cloth containing
42 yards for iS22 105., of which he sells 27 yards for
JS15 155. ; how many yards has he left, and what have
they cost him ^ A7is. 15 yards ; and they cost him
£6 15s.
34 A merchant bought 234 tons, 17 cwt., 1 quarter,
23 lb, and sold 147 tons, 18 cwt., 2 quarters, 24 lb ; how
much remained unsold ? Ans. 86 tons, 18 cwt., 2 qrs.
27 lb.
35. If from a piece of cloth containing 496 yards, 3
quarters, and 3 nails, I cut 247 yards, 2 quarters, 2 nails,
what is the length of the remainder } Ans. 249 yards,
1 quarter, 1 nail.
36. A field contains 769 acres, 3 roods, and 20 perches,
of which 576 acres, 2 roods, 23 perches are tilled ; how
much remains untilled ? Ans. 193 acres, 37 perches.
37. I owed my friend a bill of £76 16s. 9id. out of
which I paid £od 17s. lO^d. ; how much remained due >
Ans. £1Q 18s. 10^d.
n
■"
126
COMPOUND MULTli'LICATION.
38. A norchant bought 000 salt ox liidcs, wcigliln;;
r>01 cwt.,2 lb; of which ho sold 2r)0 hides, Vei«,'hiu3
2.39 cwt., 3 qrs., 25 lb. How many hides had ho left,
and what did they wci^^h ? Ans. 350 hidoH, woighin<r
321 cwt., 5 lb.
30. A merchant has 200 casks of butter, wcifdiing
400 cwt., 2 qrs., 14 lb; and ships off 173 c"ask,s,
weighing 213 cwt., 2 qrs., 27 lb. How many casks has
ho left; and what is their weiglit .> Anx. 36 cuska,
weighing 186 cwt., 3 qrs., 15 lb.
40. What is the difference between 47 ]^]ngHsh miles,
the length of the Claudia, a Roman aquoducjt, and 1000
feet, the length of that across tiie J^ee and Vale of
Llangollen ? Ans. 247160 feet, or 46 miles, 4280 feet.
41. What is the difference between 980 feet, the
width of the single arch of a wooden bridge erected at
St Petersburg, and that over the Schuylkill, at Phila
delphia, 113 yarda and 1 foot in span .? Ans. 640 feet
QUESTIONS FOR Til 12 PUPIL.
1. What is the rule for compound subtraction ? [16].
2. How is compound subtraction proved ? [19].
COMPOUND MULTIPLICATION.
20. Since we cannot multiply pounds, &c., by pounds,
&c., the multiplier must, in compound multiplication,
be an abstract number.
21. When the multiplier doos not exceed 12 —
Rule— I. Place the multiplier to the right hand
side of the multiplicand, and beneath it.
II. Put a separating line under both.
III. Multiply each denomination of the multiplicand
by the multiplier, beginning at the right hand side.
IV. For every time the number required to mako
one of the next denomination is contained in any pro
duct of the multiplier and a denomination of the multi
plicand, carry one to the next product, and set down the
remainder (if there is any, after subtracting the number
equivalent to what is carried) under the denomination
i
COMPOUND MULTIPUCATIOX.
127
wcigliln;*
ho left,
wuighing
wciglung
3 casks,
asks has
6 casks,
>h miles,
nd 1000
Vale of
280 foot,
'oet, tho
cctcd at
t Phila
40 foot
? [16].
pounds,
lication,
t Land
iplicand
e.
3 mako
ny pro
multi
)wn the
nurnhfir
liuation
to whicjh it bolouf^s ; hut should then; bo lu) remaiudcr,
put a cypnov in that duuouiination of the protluct.
22. Example.— Multiply £62 17s\ lOd. by 6.
X 5. d.
C2 17 10, multiplicand.
0, multiplier.
377 7 0, product.
Six times 10 ponce are 60 ponce ; these aro equal to '>
phillinpfs (5 times 12 ponce) to no carried, and no pence to
be sot down in tho product — wo therefore write a cypher in
tho pence place of the product. 6 times 7 aro 42 shillingM,
and the 5 to be carried are 47 Bhillings— wo put down 7 in
the units' place of shillings, and carry 4 tons of shiUings.
6 times 1 (ten shillings^ aro G (tons of 'lillings), and 4 (ten.s
of shillings) to be carried, aro 10 (tons of shillings), or 5
pounds (5 times 2 tens of shillings) to bo carried, and
nothing, (no ten of shillings) to bo set down. 6 times 2 pounds
are 12, and 5 to be carried are 17 pounds— or 1 (ton pounds)
to be carried, and 7 (units of pounds) to bo set down.
6 times 6 ('tens of pounds) are 30, and 1 to bo carried aro
37 (tens or pounds).
2o. The reasons of tho rule will be very easily understood
from what we have already said [Sec. II. 41]. But since, in
compouud multiplication, the value of tho multiplier has Jio
cotinexion with its position in reference to tlic multiplicand,
whore we set it down is a mere matter of convenience ; neither
is it 80 necessary to put cyphers in the product in those deno
minations in which there aro no significant figures, " ^ it is in
flimplo multiplication.
24. Compound multiplication may be proved by re
ducing tho product to its lowest denomination, dividing
by the multiplier, and then reducing the quotient
Example.— Multiply £4 3s. Sd. by 7.
£> s. d. Proof :
4 3 8 29 5 8
7 20
29 5 8, product.
585
12
7 )7028 , product reduced.
12 )1004
io)ou e
quotient reduced "~4 3 8=lnultiplicand.
123
COMPOUND MULTIPLICATION.
pm«'i;„'trri''hUy performed" tr"' "•,"'«■"«'«, the
»l.ouM be equal to ,T,e ICS ' " '°™"' '"'" "'' ">''
ihe quantities are to bo " rednrpfl " hoft^r.« tu^A , 
gnce^the^earuer i, .ot supp'^tit' be'S :%tT"a^iJe
1.
2.
3.
4.
6.
6.
7.
8.
9.
s.
9
2
18
7
13
1
8
12
17
6
d.
3.
7i
6.
li.
U.
Oi
0.
6.
6.
8.
0.
EXERCISES.
£ S. d. £
76 14 7iX 2= 163
97 13 6iX 3= 293
77 10 74X 4= 310
96 11 7iX 6=482
77 14 6ix 6= 466
147 13 3^X 7=1033
428 12 7iX 8=3429
572 16 6 X 9=6155
,A ii^ ^'^ ^ X 10=4288
10. 672 14 4 Xll=7399
W ir'l l^ ^ X12=9321 u u.
12. 7 lb at 5*. 2M #•, will cost £1 16s. Hd.
14* ?/l'ii' ""^ W^i*^ ^' ^"1 «<^«* ^4 18,. 5J,i.
ii'J^ra\7ii3^t.^^hrs^^^^^^^^^
Rule.— Multiply successively by its factors
Example L— Multiply £47 13s. U. by 56
£ s. d. ^ '
47 13 4
50=7x8 £ ,, ^
333 13 4=47 13 4x7.
8
T
9
E
Til
[8ec,
tlie 1
mult
2669 6 8=47 13 4x7x8, or 56.
Example 2.— Multiply 14^. 2d. by 100.
s. d.
14 2
100=10x10 ,. ,;.
£7 1 8=14 2x10.
10
X70 16 8=14 2x10x10, or 100.
COMPOUND MULTIPUCATION.
ExAMPip: 3.— Multiply ^8 2s. 4c/. by 700.
£> s. d.
8 2 4
10
— £
129
81
3 4 =8
10
811 13 4 =8
7
s. d.
2 4x10.
2 4x10x10, or 100.
568113 4=8 2 4x10x10x7, or 700.
The reason of this rule has been already given [See. II. 60].
26. When tlie multiplier is the sum of composite
numbers —
Rule.— Multiply by each, and add the results.
ExAMPLE.^Multiply £3 Us. M. by 430.
£ .9. d.
3 14 6
10
£ s.
37 5 x3=lll 15
10
d. £ s.
0, or 3 14
d.
Cx30.
372 10 0x4=:H00 0, or 3 14 6x400.
IGOI 15 0, or 3 14 6x430.
r^'^^'V^nn" ?m *''^ ''""^^ ^'^ "'® «"'"° "« t^'^^t «li'e«<^y given
[bee. II 52]. Ihe sum of the products of the multiplicand hv
the parts of the multiplier, being equal to the product of the
iuulti23hcand by the whole multiplier.
EXERCISES.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
on
27.
28.
£
3
4
6
2
3
2
3
•s.
7
16
14
17
16
3
4
9
16
d.
6 X
7 X
6iX
6 X
£
18= GO
20= 96
22=125
36=103
66=214
64=139
81=261
100= 46
X 1000=816
100 yards at 9*. i^d. ^W, will cost £46 17
oVa S*i^'*^ »' ^•^* *'f 4K, will cost 466 13
4 gallons at (!.?. 8./. .W, will cost 80
bbO yards at 13a. 4c/. 4(^', will cost 240
X
X
X
X
s.
15
11
19
10
8
4
11
13
13
d.
0.
8.
11.
0.
8.
0.
«.)
".
4.
4
6.
4.
0.
0.
130
COMPOUND MULTIPLICATION.
27 If the multiplier is not a composite nun^ci*';'^
lluLE. — Multiply successively by the factors of tho
nearest composite, and add to or subtract from the pro
duct so many times the multiplicand as the assumed
composite number is less, or greater than the giv9i\
multiplier.
Example 1 —Multiply £G2 12s. Od. by 70.
£ s. (L
62 12 6
8
76=8x9+4
501
9
£ s. (I.
4509 0=G2 12 0x8x9, or 72.
250 10 0=02 12 0x4.
4759 10 0=62 12 0x8x9+4, or 70.
Example 2.— Multiply £42 3s. 4(L by 27.
£ s. (I.
42 3 4
4
27=4x71
108 13 4
£ s.
1180 13 4=42 3
42 3 4=42 3
d.
4x4x7, or 28.
4x1.
1138 10 0=42 3 4x4x71, or 27.
The reason of the rule ia the same as that already given
[Sec. II. 61]. ^ ^
EXERCISES.
£ s. d. £ s. d.
29. 12 2 4 X 83= 1005 13 8.
80. 15 0ixl46= 2193 3 Oi
31. 122 6 X102= 12469 10 0.
82. 963 0^X999—962040 2 5i.
28. When the multiplier is large, we may often con
reniently proceed as follows —
Rule.— Write once, ten times, Sic, the multiplicand,
&c., of the multiplier, add tho results.
COMPOUND MULTIPLICATIO^f.
131
Example.— Multiply £47 16s. 2d. by 5783.
5783=5 X 1000 + 7 X 100+8 x 10 + 3 x 1.
£> •«. d. £ s. d.
Units of the multiplicand, 47 10 2x3= 143 8 6
10
Tens of the multiplicand, 478 1 8x8 =
10
Hundreds of the multiplicand, 4780 16 8 X 7 =
10
3824 13 4.
33465 16 8.
Tliousands of the multii.Iicand, 47808 6 8x5 = 239041 13 4.
Product of multiplicand and multiplier =27647.5 11 10.
EXERCISES.
33. 76 14
84. 974 14
85. 780 17
d. £ .9. (I.
4 X 92= 7057 18 8.
2 X 76 = 74077 16 8.
4 X 92=71889 14 8.
7ix 122= 9013 10 3.
7ix 162= 6865 11 lOJ.
38. 76 gallon.^ at £0 13 4 4f , will cost £50 13
39, 92 gallons at 14 2 4f , will cost 65 3
36
37
73 17
42 7
4.
4.
40. What is the difference between the price of 743
ounces of gold at £3 17s. lO^d. per oz. Troy, and that
of the same weight of silver at 62d. per oz. .? Ans.
£2701 2s. 3^d.
41. In the time of King Jolm (money being then more
valuable than at present) the price, per day, of a cart
with three horses was fixed at 1^. 2d. ; what would be
the hue of such a cart for 272 days ? Ans. £15 175. 4d.
42. Veils have been made of the silk of caterpillars,
a square yard of which would weigh about 4 grains ;
what would be the weight of so many square yards of
this texture as would cover a square English mile >
Ans. 2151 tb, 1 oz., 6 dwt., IC grs., Troy.
QUESTIONS TO BE ANSWERED 13Y THE PUPIL.
1. Can the multiplier bo an applicato number ? [20J.
2. What is the rule for compound multiplication
when the multiplier does not exceed 12 ? [211.
3. What is the rule when it exceeds ^12^ and is a
composite number ? [25] .
m
i!li
132
COMPOUND DIVISION.
4. When it is the feum of composite numbers ? [261
6 When It exceeds 12, and not a composite number ?
6. How is compound multiplication proved ? [24].
COMPOUND DIVISION.
29. Compound Division enables us, if we divide an
apphcate number mto any number of equal parts, to
asceitam what each of them will be; or to find out
anrth'^r^"^ ^'"'''^ """^ applicate number is contained in
If the divisor be an applicate, the quotient will bo an
abstract number— fur the quotient, when multiplied by
the divisor, must give the dividend [Sec. II 7yl • but
ronV^^^ri"!!' ?>'^^''' ''^"'"^* ^« multiplied together
L20J. If the divisor be abstract, the quotient will be
applicate— for, multiplied by the quotient, it must give
*he dividendan applicate number. Therefore, either
tlivisor or quotient must be abstract.
ceeTl^^^"^ *^'^ ^^^'^'^'' ^^ ''^^*''^«^' ^^^ <ioes not ex
RuLE— I. Set down the dividend, divisor, and sepa
rating Ime— as directed in simple division [Sec. II. 72].
II. Divide the divisor, successively, into all the deno
minations of the dividend, beginning with the highest
111 Put the number expressing how often thl divisor
s contamed in each denomination of the dividend under
that denomination— and in the quotient
tinnTf tL* r •^''''',°' ''^'^^J contained in a denomina
tion of the dividend, multiply that denomination by tho
number which expresses how many of the next lower
denommation is contained in one of its units, and Tdd
the product to that next, lower in the dividend.
wn V * ..A^^^^l ^""^^ succeeding remainder in the same
way, and add the product to the next lower denomi
nation in the dividend. umomi
VI. If any thing is left aftor thn nnotint ^"o ^i
lowest denomination of the dividend is obtained; pi^t'iJ
COMPOUND DIVISION.
13:
in
I
down, with the divisor under it, and a separating lino
between : — or omit it, and if it is not less than half
the divisor, add unity to tho lowest denomination of tlie
quotient.
'61. Example 1.— Divide X72 6s. did. by 5.
£> s. d. '
5)72 6
14 9
4i
5 will go into 7 (tens of pounds) once (ten times), ami
leave 2 tens. 5 will go into 22 (units of pounds) 4 tiraos. and
leave two pounds or 405. 405. and (Ss. are 4Gs., into which 5
will go times, and leave one shilling, or lid. 12t/. and \)d.
are 21tZ., into which 5 will go 4 times, and leave Ir/., or 4
farthings. 4 farthings and 2 farthings are 6 farthings, into
which 5 will go once, and leave 1 farthing— still to be divided ;
this would give \, or the fifth part of a farthing as quotient,
which, being less than half the divisor, may be neglected.
A knowledge of fractions will hereafter enable us to
understand better the nature of these remainders.
Example 2.— Divide £52 4s. l^d. by 7.
& s. d.
7)52 4
n
7 9 2
One shilling or \2d. are left after dividing the shillings,
which, with the Id. already in the dividend, make 18(7. 7
goes into 13 once, and leaves 6rf., or 24 farthings, which,
with f , make 27 farthings. 7 goes into 27 3 times and G
over ; but as G is more than the half of 7, it may be consi
dered, with but little inaccuracy, as 7— which will add one
farthing to the quotient, making it 4 farthings, or one to
be added to the pence.
32. This rule, and the reasons of it, are substantially the
same as those already given [Sec. II. 72 and 77]. The remain
der, after dividing the farthings, may, from its insignificance,
be neglected, if it is not greater than half the divisor. If it is
greater, it is evidently more accurate to consider it as giving
one farthing to the quotient, than 0, and therefore it is proper
to add a farthing to the quotient. If it is exactly half tho
divisor, we may consider it as equal either to the divisor, or 0.
33. Compound division may be proved by multipli
cation — since the product of the quotient and divisor,
plus the remainder, ought to bo equal to the dividend
[Sec. TI. 79].
g2
134
COMPOUND DIVISION.
EXERCrsKS.
1.
o
90
7
7(5 14
47 17
10
11
3.
4.
6.
6.
7.
8. 97 14
9. 147 14
167 16
176 14
96 19
77 10
32 12
44 16
d. £
61. 2=48
7j y=25
0^ 4=U
4: 5=19
0=12
7=4
7i
o
ft. d.
3 9.
11 6i.
19 U.
7 10.^.
19 5i.
13 '3.
7 f. 8= 5 12 l"
17 1.
16 5i
3^ 9=10
0410=14
7Ml=14
6j12=14
6
14
11)
6i
The above quotients are true to the nearest fa. dnl^.
number!?''' '^' ^^"^^''^ "^'^^^^ ^^' ^"^ ^^ ^ composit.
Rule.— Divide successively by the factors.
ExAMPLt.Divide £12 175. U. by 36
3)12 17 9
This rule will be understood from Sec. II 97.
12.
EXERCISES
£> i. d. £ ^.
J* II 6^ 24= 1
d.
81
13. 676 13 34 36=16 4]
14. 447 12 25. 48= 9 6 6
16. 647 12 4^ 60= 9 i? 7
17. 740 13 45 49=15 2 sj.
Jte n^W ^^"^ '^'"''^ 12, and ia no. a oom
h„fi!i!''^7^™'"""^, ""y *''<> ""ttod of Ions division •
set down tlie inultiplica, fco obtinM "' «"'aond,
tient as directed in long'dH.ision 'tSo II.'S]! '^'""
■"^
iposite
lom
lon ;
tiers
eno
iiext
[)nd,
j[uo
t;OMP0U.\D DIVISION. 135
ExAMPLK.— Divide £87 IG.v. 4d. hy G2.
£> s. (I. £, ,<. f/.
62)87 10 4 (1 8 4.
G2
25
20 multiplier,
shillings 5 16 (=25x20410)
490
"20
12 niultiplior.
pence 244(=20xl2f4)
186
~58
4 multiplier
farthings 232 (=58x4)
180
"40
C2 goes into £87 once (that is, if gives £1 in the nuoti.nir),
ard leaves £25. £25 are equal to 500.s\ (25x20j, Avhich,
with 10.S. in the dividend, make SlCs. 02 goes into 51fis'. 8
times (that is, it gives 85. in tlie quotient), and leaves 2(»i.,
ov 240^2. (20x12) as remainder. 02 goes into 240, &:c.
Were avo to put I in the quotient, the remainder would ba
40, which is more than half the divisor; we consider tlm
quotioiit, therefore, as 4 farthings, that is, we add one penn^
to (3) the pence supposed to be already in the quotient.
£1 8*'. Ad. is nearer to the true quotieut than £1 8s. 34Vi.[32].
This is the same in principle as tlie rnle given above [30]—
buv since the numbers are large, it is more convenient actually
to set down the suras of the dilVorent denominations of the divi
dend and the preceding remainders (reduced), the products oJ
the divisor and quotients, and the numbers bv which we multi
ply for the necessary reductions : this preveuta the memory
from being too much burdened [Sec. II. 93].
36. When the divisor and dividend are both applieate
numbers of one and the same denomination and no
reduction is required —
FtULE. — rroeeud as already dii'ected fScc. II 70,
72, or SU]. '
136
COMPOUND DIVISION. 
Example.— Divido £45 by £5
£5)45
Tliat is £5 is the ninth part of £45
nominatiou is found in cither, oi' both^ '"' ^'•
ItULE.Ileduco both divisor and dividend M fl.n i
est denomination contained in eithor m / .u^ ^^"^
ceed with the division ^^^' ^^^ *^^^^ P^^
ExAMPLE.Divide £37 5.. 9ld. by 3.. Ghl
'' i'. £ s. d. ' '
3
12
"42
4
170 farthings.
5 9J
M
170)35797(211
340
179
170
"~97
Theroforo 3.. Qid. i, (ho
211th part of je37 5y.9,V/.
*J7 not being less than the half of 170 ^91 ^^^ i .
OS equal to the divisor, and therefore addSf J fT Tr''^?'' '*
as the last quotient. ^"^^eioie add 1 to the obtained
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28
29
SO
£ s.
176 12
134 17
1736
14
73
16
147
14
157
IQ
68
15
62
10
8764 4
4728 1;}
8204
5236 2
EXEHCISES.
d.
2 ^
8 ~
7 ^
7 f
s ^
7 1
2 i.
6i^
Oi~
2 f
Hi
7^
2 —9842:
• 191:
183:
443=
271:
973=
487=
751 =
419=
408=
317=
261:=
875=
£ s.
■ 18
d.
6.
14 9.
10 13 104
" 6
3
6
1
2
5i.
5.
=
=
=
:
:18 14
:14 18
31 10 114.
■■ 5 19 81
9 4].
111.
Gh.
41
COMPOUND DIVISION.
137
licato, but
a one de
) the loT?
then pn.
31 . A cubic foot of distilled water weighs 1000 ounces
what will be the weight of one cubic inch ? Ans
253'lS2i) grains, nearly.
32. How many Sabbath days' journeys (each 1155
yards) in the Jewish days' journey, which was equal to
ii3 miles and 2 furlongs English > Ans. 5066, &c.
33. How many pounds of butter at ll^d. per lb
<vould purchase" a cow, the price of which is £14 15s. ?
Am. 3012766.
QUESTIONS FOR THE PUPIL.
1. What is the use of compound division .? [29].
2. What kind is the quotient when the divisor is an
a])stract, and what kind is it when the divisor is au
applicate number ? [29] .
3. What are the rules when the divisor is abstract,
and docs not exceed 12 ? [30] ;
4. When it exceeds 12, and is composite ? [34J ;
5. When it exceeds 12, and is not composite ? [35] ;
6. And when the divisor is au applicate number ? [36
and 37].
'■ i^ lh(
isidor it
)btainoJ
13S
Sii:oTioN jy.
FRACTIONS.
?n<l one TZZ'^CtZlf^'f.''''" ^I""' P"'«.
is called a /„„«« ""''° P«'« »'•'> token, we have what
re,t';4:r;';:;'Lt3eSt'%'''!,.'' " been
la a fraction of 5— its siv(l, 1 . !i •'^ '' f^"" H ««!)
into six e,,„al part f w n ?tn ' *""" ""'Z boingdivid i
wo shall sJo prLen iy) »' un?^^™?,^ *'""» i "' (»»
4 Srchtgo^fcl^ZV"^^^^^^^^^^ » fr
or, while the Jeuom „'uoV tell the ^' ''"■mimtor ,■
kinci of parts into which the nn;, *"'"'"««*« or
divided, the numerator "m„!t/« J, '"^f"""^ '» ""o
number of them which TtTn t!"' "l f*'"""'^ 'l^
sevenths) means thottL ! ,/''"M (read three
" 'tree V"dr'ttXm''ed ^^r''^'" ""^ "'
'Trei;::?;t:'/» ^~th?f::r:.» ^
of the fraot! nX~h™ .'• '^^ ?'°'" ""^ ™'"
JiviJc the numeSor bv thT"J'™' "^'"^'^ *'"=" we
value ; and the Kreater .l,r^^ •T"]"'"''''' '' "« ■''^al
quotient. On the^eomrart .h7*",'* '¥ ''■"■§«"■ «'o
♦or the less the frae on^:,' T''^ ""^ denomina
'1 smaller the quTt „t7seo jf 7^^^^^^^ "^"^
greater than 4— whieh ;« i? i T®^ =— hence 4 is
nun.r'atrr^ ?dXls^r' V'''^^«' "' »'
^l«'ineha,^^d.1l^X;r^lSn"S
• ^ xro ean
no
FRACTIONS.
139
to increaso or <liinini«U both the (Ji\ iJciiJ and divisor —
which does not afFoct the (diotieut.
5. The following will rc^. resent unity, sevensevenths,
and livesevenths.
I
I U"^^y I Lii
?i
The very faint lines indicate wliat ^ wants to make
it equal to unity, and idmlical with ^. In the diagrams
which are to follow, we shall, in this manner, generally
subjoin the difference between the fraction and unity.
The teacher should impress on tlie mind of the pupil
that he might have chosen any other unity to exemplify
the nature of a fraction.
6. The following will show that ^ may be considered
as either the f of 1, or the  of 5, both — though not
identical — being perfectly equal.
\ of 5 units.
? of 1 unit.
Unity,
DIDIl
m
1
\fi\f
o\
Tn the one case we may suppose that the five parts
belong to but one unit ; in the other, that each of the
five belongs to differeii ' units of the same kind.
^ Lastly, ^ may be considered as the  of one unit fivo
times ua large as the former ; thus
I of 1 unit.
I of 5 units.
equal to
■I ..
.1 . .
0\
"•*
HO
FnACTIONS.
at leapt, of the quotiont^nUo'Llntotr '"'''''•' ''^'''
not. uknHcalZk''Z , j^lf C.S'T*"V!"' ^'
lowmff will exomnlifv +].! • "'^^"^" h iho foi
thus J i3 evidently eqS to 'r"*' *f f »''■">»"*»■■ ;
noticed when we trite'd of divisfo^ Lril.Tl/'"^"^
tbo>;e4eteabt";7t\"n;tL"° '"''™ ".^Z^''
KKACTION«.  I4j
11. A cnmpnwnd fraction siipposos ono fraotioa (o
refer to another ; thus J of J— roproscntiid iilso hy J X ^i
(throefourths multiplied by fourninths), means not
tho fourninths of unity, but the fournintlia of tho
throofourths of unity :— that is, unity boia,^' divided into
fjur parts, tlirco of thcso arc to bo divided into nino
parts, and then four of those nino arc to bo taken ; thua
I ?
o
TIT
I r
12. A complex fraction has a fraction, or a mixed
number in its numerator, denominator, or both : thus t,
4
which means that wo are to take the fourth part, not
of unity, but of the ^ of unity. Tiiis will bo eicm
pufied by —
8 ? U U

3
r
1
f<:cj^
J
1
^
u
7' s'
1 4
5
— , are complex fractions, and will be better
54
understood when we treat of the division of fractions
13. Fractions are also distinguished by the nature of
thou' donommators. When the denominator is uniiv,
foUowed by one or more cyphers, it is a decimal frac
iion—imL^, /it, f Ao , &c. ; all other fractions are viloar
^thus,i,,73_, &c.
Arithmetical processes may often be performed with
fractions, without adnalhj dividing the numerators by
the denominators. Since a fraction, like an integer
may be increased or diminished, it is capable of uJli*
tion, subtraction, &c.
142
FRACTIONS
niinaticm!' '''^'^' """ "'^'°"' ^° '' ^''''^^^'" '^" ^^^ ^^"«
uni^v"" !r.^T' "^^ ^ co^'^idercd as a fraction if wc make
unity it^ denominator .—thus f may be taieu for 6 •
if 21° '""^ ^7" ™, '."J^S"! any denominator we please
If we previously mnltiply it by that denominator ;
tims, 5=1 , or f , or f , &c., for 22=5X5_5_5 .
, 30 5X6 5 ' ^ 1X515,
and — = ==5 &«
6 IXQ I ^
EXERCISES.
^i S^'^''''^ '^ *° ^ ^'■''''*'''°' ^^""'"S ^ ^' denominator
tor^" I'f."' o^^ *° ^ ^'^'*^'"' ^^"^^"S ^^ ^« denomina
^',i^^'' l^^' 1?=V. I 5. 42= V/. I 6. 71 = «F«
lo. lo reduce fractions to lower terms
Before the addition, &c., of fractions, it will be often
Trirplpe ^^ *'^^^ *^^"^^ ^^ ^^^<^^  P^^^
■n, 40 5 40 40i8 <!
Example.— =j^=. For ~— 3^^Xr_^
72 9 '^^'7272^89•
We have already seen that we do not alter the quotient
which 18 the real value of the fraction [4]if we XSv ot
divide the numerator and denominatoh)y the Tame number
he^atr:' ^''" ''''' ^'' "• ''"' ^^^^ ^^ usefull^rem , :
Reduce the
7 574 387
10. MJ%
12 ?§CFr'
In the answers to
in future, generally
mlnations.
EXERCISES.
following to their lowest terms.
13.
14.
15.
16.
17.
18.
Fa 3 1 •
4 3 4
Fll 3"*
60 5
12 S"
.98 7
1 1 2 — y
19,
20,
21.
22.
23.
24.
100400 1004
7^00 2'
5UO0 1120
firnj — t:?ii'
4 2 5__ s's" '
1^^, — I?T'
30 8
4 12.
"BTr="2n3
5I2_2 5
B J i — SoT
questions given aa exercises, we shall,
reduce fractiona in fhoJi. In^nof d Z_
FRACTION*.
143
ny deno
we make
I for 5;
e please
5
nmator
lomina
8514
»4 '
16 often
ossible,
ommon
tient—
tiply or
umber,
•emein
— 1004
3ffoa
1 20
?T1'
h'
shall,
^ —
28. £'=ir).v.
29. £j^=5.s.
30. £^„=1./.
1«.'^ find the value of a fiactiou in terms of a
lower a6iaomination —
RuLR.^;^rteduce the numerator by the rule already
given [Sec. III. 3], and place the denominator under it.
ExAMPLK.— What is the value, in shilUngs, of J of a pound ^
£3 reduced to 8hillings=60.s. ; therefore £'i reduced to shil
liiigs=';j^s.
TJie reason of the rule is the same as that already given
[Sec. III. 4]. The  of a pound becomes 20 times as much if
the " unit of comparison" is changed from a pound to a shilling.
We may, if we please, obtain the value of the result
ing fraction by actually performing the division [91 •
thus \°s.~los. :— hence £^ = l5s. *
EXERCISK.S,
25. £U=Us. 6d.
2G. £1^=1 7s. 4d.
27. ^ll^lOs.
17. To express one quantity as the fraction of an
other —
KuLE. — Ecduce both quantities to tho lowest deno
mination contained in either — if they are not already
of the same denomination ; and then put that which is
to be the fraction of the other as numerator, and tho
remaining quantity as denominator.
Example.— What fraction of a pound is 2W. ? £1=9G0
earthings, and 2,./.=9 farthings; therefore ^fr^ is the re
quired fraction, that is, 2^(1. =£y^^.
om '^•^Tt^ ^*' "^^""^ lluLE.— One pound, for example, contains
JfM fartlnngs, therefore one farthing is £.' (the OGOth part
of a pound), and 9 times this, or 21, is £9 X yo=ffT!r
EXERCISES.
31. What fraction of a pound is 14?. 6d. ? Ans. ^a
32. What fraction of £100 is 17^. 4d. ? Ans. ia°."
33. What fraction of i^lOO is i:32 10^. ? Jm.*i!
34. What fraction of 9 yards, 2 quarters is 7 yar*ds,
3 quarters ? Ans. .ai. *^ '
35. What part of an Irish is an li^nglish mile > Ans n
36. What fraction of 6s. iid. is 2s. Id. ? A,is. X '
ov. What part of a pound avoirdupoise is a pound
Troy? Am. m. ^
u^
«.'
VULGAR FRACTIONS
QUESTIONS.
1. What is a fraction ? [1]
.. 7/ >Vhj may the numerator and denomhiatnr he. «,„i
o. What IS an improper fraction ? [71
b. What IS a mixed number ? [8]
numbejTrg]"" ""^^P^' ^'■^'"^"^ '"^uccd to a maed
denomination? [14]. ^ *^ "^ fraction of any
^^12. How is a fraction reduced to a lower term?
a ^Jt2^^^^ ^^ ^^^  terms of
of ItotfeT' tl7T ''^'''' '"' ^"'"'^^^ ^« the fraction
VULGAR FRACTfOiNS.
ADDITION.
18. If the fractions to ho Tddo/I i,n,,^
denominator— ^^ ^^'^^° ^ common
Example. — a 4. g — i t
Reason of THF Tfrrrw T^ I,
if
VULGAR FRACTIONS.
145
tbeir nature.
Unity.
1. 3K+^=v==i5.
3. .'4'"4.iL=;!o_o4
O. f ^U'*_l_lo 4 2 03
EXEnciSES.
I t), l^J_ll_l_l«_33_0
10. lT4.'t!f.iii_.5_^_.oin
12. ^\f a 4.11^3^^1 ri?
19. If tlie fractions to bo added Iiave not a'c'ornni..n
a«ator, and all the denonnnato.s are p iL^rS
Example.— What is the sum of 5f 34.4 ?
+^+ ^=.2X4X7 3x3x_7 4x3xi ^6 63 48 m
H • ;^t^'^X^X^^7x3x4=84+84+84=8T
factors^ (tlio ' v^Mdenomnnt T"' "''/"'' ^ ''^"«« ^'^« ^=""«
the same product '^'"""^^'^^^^^ors) mu«t uecemrilj produce
confmof trmrn'aoT we^'w""''' ^^"^ "'^ ^^^^'^ *« a
rator and denomi air of e^^ "Ct/^^'^'P"''^ ."'« ""'"«'
[4] does not alter the fraction "^ Tf '^'"^ ""•"^'''•' ^^'^<^^
common denominator; for 7f L „ ],/%r''T'''^: ^^ ^""^ <*
out so doin,, w. cannot pil ^1^ tloililaJTri^^n^it
of them as tlie denominator of tlieir sum;thus ^^±^+^
So ^uSffi^r t jiii^ns^r .:? r ^ ^^'^ ' ^^^
and sevenths. Avhich avo Lo! 1, J'"''P."'^ ^''^"» "»« <onr(l,3
N^+i
are less t)
be correct —
tl.an thirds; noiihor wuukt
«ince it would suppose nJi ,<y tU«m to be
f
146
! 1 Tilt ,
lis
VULGAR FRACTIONS.
equivalent number of others wl.icl L"e ,„,„ll,r 1 ' '° "°
H+?=2_X4 3X3__8 9 17
_ 3 4 8X4'^4X3~12+12"=J2
These fractions, before and after thev r..r.fMvn ..
denommator, will be represented as follo'y^s : '"""'^
8
1
('
equal to
■ w iai n an^JL *—
as w: ilr diss ihd/^i^^^f °' ^'^^ r? j^  '«
than twelfths, we eouk not C« f ''!,"*'^ ^"'^'" I^'"'^'^ l'"^'««
exactl,oquivalenl"res;!ectitely:iot?AS^r '' '''"'
12 3+f+f
EXERCISES.
20
=5 9 139
22. If the
denomiuutor,
to each other
Procoed as
Rule. — Fi
denominators
fractions to be added have not a common
and all the deuommators are not primo
directed by the hist rule ; or—
"d the least common multiple of all the
[Soc.II.l07,&c.,thiswilllieth.cnnnnnn
muifciply tno numerator of each fraction
8n_i 5
i coniiiiou
VULGAR FRACTIONS. ^4*
into the quotient obtained on dividing the common mul
tiple by Its donominatorthis will give the new nume
vators ; then add the numerators as already directed [18].
ExAiMPLE. — Add X I A I 3_ OQQ • .1 ,
■^^^ 3 2 14^5 f^ 2 Z6ii IS the least common
multiple of 82, 48, and 72 ; therefore i.l.i_288i82x5
.281f:48X4 288^72X8 45 24 ' if 8^"""^^^""
ii88 + 288 =288+2T8+288=2T8
multiple of the denomiLts^K^Tnt ^6^si'JT''''
__ 5><288 ^ ^^""^
instance) g^q^. For we obtain the same quotient, whether
diminish the numher^f^^^^^^^^^^
tionsTe oS.lr h^'''''^'"' ""? ""'^ P""^° *'' «^«I^ other the frac
jS>?e ff ^e a:::^:z:::^i^j^t^]^^:^
denommators, the common denominator irthe present n
stance, had wo proceeded according to tho L, ru^e ?9], wi
would have found 1 , j. , j. __ 17280 18432 4608
4032£ . ^^^ 40320 ^^^^"^2110592+110592+110692=*
110692 ■ " 110592 ^^ evidently a fraction containing larger
terms than .
288"
25. ^_1_SI4.^14 3 O'JS
2b. 44.^_Lj__t;7 9."
27 ?TfTf >2— r2.
EXERCISES.
^45 i 4_.:jii.3__24 7
34. ;v?+=fCl^
35. flflL45i?''
37. +HJ!f!fe:Z:?;T^7
»2T0"'
28. l+lLi^^dllZ^Vl^
29. iJiJ niiST^^
30. It^=,i?Z{y
tioni^'" ''^'''" ' ""^^'^ ^^"^^^^ to"iu" improper frac
ha^n.''';il^!?"^%*^° '^^'^'''^ ^'^'^ i«i« a fraction,
te.^:dn;tri^L.^,.r ^^^ p
^^Ex^PLK.What fraction is equal to4f ? 4^ = 4_f.5=,
Mi
II '.
148
VfLUAU FRACTIONS.
o. Reason of thk Rirr p _ \Va i „ i •.
integer may bo expre;sc. us a ^r u Hon f '"'''^ '''^ '^'^' ^
nator >ve pksase :the reduction nf„ •'"'T'''" ''"^ ^^'"^""
fore, is really the addition of Z.r '"'''''' ""mhor, tliere
a coiiiiuou deuomiaator. ^^''^"tions, previously reduced to
EXERCISES.
44. 99A.=:iono
45. 12''.i=>i^ •
40. i4Lo/: ■
47. 40 '==3 73
48. 13 =1 o;
49. 27=iv.
38. 16'=''3^
39. 18=io.
40. 79.=«3.
41. 47 ='3.
42. 741=017.
43. 95^=^f6.
26. To add mixed numbers—
KuLE.Add together the fractional parts th^n if
«ho sura IS an imnroner fr^ot;..^ T P. ^^' ^'len, if
■="»"■« 1What is the sum of 41 + 18 "
7 I 5 12 14
«T^8 — 8 =Ji
sum
isl
eighthsthat ; , one o^t a;Hed''^n;: .**; T '"''' ^'^ ^
and 18 are 19, aid 4 are 23 ' "" "'*^ ^"'^" ^
KsAMPLE 2.— Add 12f and 29i.l.
4==4l=l.i 7 12^=12.3 5
3
sum 42'I
y
T„ .1 • . . sum 4z.'ri
rW an,r 22f t'XTS l^fj H'''""""*? ""^ Edition
aenominatol^ ""* ''^"'"""^ l^to to ,v oomiuon
beSis"ptSir,?er™'il"*"'» ^' "'0 '™
but, in the first exa™t.efi,ril?''''? f ""P'" "M'ion
Won is equal to „S oTthl „" . ST'"*^' ''.'' O"" <>™"mi„a.
.1. 8], J« Of one dealttr„'^*!S '^ TeTf ttt"." ''^°
50. «+3=8S. '"^"'■'"='
61. 8f4.4.2l==lltRi
52. l65'HfOG^L^"'
53. 10+11^.^22^
54.
II MS
56. 4r+3§j:H6 =n »
58. 92.4.+37Ji+7tifi7J5,
59. 17aA+8!3tlT=27^J.
VULGAR FRACTIONS.
149
th.it an
dcnomi
'', tliere
(iuced to
if
set
QUESTIONS.
1. Wliat is tho rule for adding fractions which have
a common denominator? f"18].
2. liow are fractious brought to a common denomi
nator ? [19 and 22].
1 ^' S** ^^ *^^ ^"^^ ^^^' addition when the fractions
have different denominators, all prime to each other ?
_lc)J.
4. What is tho rule when tho denominators are not
the same, but are not all prime to each other ? [22].
5. How is a mixed number reduced to an improper
fraction ? [24] . ^
6. How are mixed numbers added ? [26].
SUBTRACTION.
28. To subtract fractions, when thcv have a common
ionommator —
^ .Rule.— Subtract the numerator of the subtrahend
ifrom that of the minuend, and place the common deno>.
iftimator under the difference.
Example.— Subtract J from J.
7_4_74_3
9 9~ 9 ""9'
20 IlEAsoN OF THE lluLE.— If We takc 4 individuals of anv
Kind, Irom t of the same kind, three of them will remain In
the example, we take 4 (ninths) from 7 (ninths), and 3 are loft—
wluch must be nintlis, since the pvoces.s of subtraction cannot
have changed their nature. The fullowin will exempUfy the
Bubtractiou of fractions :— a ^ ni>uiy me
^^
Unity.
7
IT
w
r
73
4
iiii
i
a>
I
H
Uil'i'
I
II i'lr
Ihif
160
VULGAR FRACnONS.
1.
2.
3.
4.
5.
EXERCISEa.
— ^,=t..
c.
7.
8.
9.
10,
1* « . 5
UXAMPLE.—Subtract ^ from .
ft! T> " " ^2 i"5=^tf.
OX. JtRASOIV OF THP T?tttw tx •
g;^ven f20] for reducing fractbns !« !''"''''' *" *'^'*^* ^^^^^'^^^
previous!; to adding them ^'^ "^ '^^'^'^o" deaominator'.
11. f
14. Ili^ll^s
'* 13 — Iffy.
EXERCISES.
_S 7
_S — 39
"8'
I
15.
16.
17.
18.
11^^13X_ 769
114 HZ's'^^**
4 8 8i — og.
75H_320_2;!
m^fd n'umberf'' '"^^^ "^°^^^' ^^ frrctions from
iVitfrsXL^^^^^^^^^^^ :." denominator
W from that of the ifnd 1'.'' ^f.^ ^' *^^ ««^*^
ence with the commS^ Zn • /'* ^^^^ *^« ^^^ff^r
subtract the integTpart ofT k'. T^'.' ^'^ *^^"
integral part of the minueL subtrahend from the
that of'Ih^:/£t nd, Clt r r '' ^ *^
mon denominator to ts nu2r./ ^ .'^'"^ *^*^^ «^°^
mtegralpartofthemi^uendT;^^^^^^^ '"' '^^^''^^ *^'
Example 1.~4 from 9.
9f minuend.
4 subtrahend.
^ /»• vxi. y. ^*^ difference.
from &'^S,4!'«''"'' d 2 eighth. (=.; ,,„^, ,
.VULGAR FRACTIONS.
151
not a coin
inator [19
rule.
at already
uominator.
7_e9
2"
ns from
inator —
subtra
e differ
!; : then
•om the
ss than
com
ise th'i
ka. 4
ExAMPLK 2.— Subtract 12.^ from 18].
18j minuend.
12^ subtrahend.
5^ difference.
3 fourths cannot be taken from 1 fmirth ; but (borrowing
one from the next denomination, considering it as 4 fourths,
and adding it to the 1 fourth) 3 fourths from 5 fourths and
2 fourths (== j) remain. 12 from 17, and 5 remain.
If the minuend is an integer, it may be considered as
a mixed number, and brought under the rule.
Example 3.— Subtract 3f from 17.
•,JJ i!?^^y ^® supposed equal to 17^; therefore 1734=3
17^3^. But, by the rule, 17^3J=163f =13^.
83. Reason of the Rule.— The principle of this rule is
the same as that already given for simple subtraction [Seo
11. ly] :— but m example 3, for iustance, five of one denomina
tion make one of the next, while in simple subtraction ten of
one, make 07ie of the next denomination.
34. If the fractional parts have not a common deno
minator —
Rule. — Bring them to a common denominator, and
then proceed as duected in the last rule.
Example 1.— Subtract 42 from 56^.
56==56y*^, minuend.
subtrahend.
42 > =
42X
I'lyV' dilTerence.
85. Reason of the Rule.— We are to subtract the dif
lerent denommations of the subtrahend from those which cor
respond m the minuend [See. II. 19]but we cannot subtract
iractions unless they have a common denominator [30].
EXERCISES.
19.
.20.
21.
22.
23.
24
157=7
12f— 12
8411 _lt_;
941
14iif— f=1473.
24. 82iH7iif=74e.
25. 76272/^^3 Jf."^
26. 6734X=32^1.
27. 97132J=64TC
28. 604ll(=19i!
1001— 9=9ni'
60— A=59,« '■
29.
30.
31.
32.
12l01=
t
u^
i'l y
I6d
TULQAR FRACnONS.
.hi
I" '^
i
QUESTIONS. <^
9 W»,o* • ^u ^^'""^on (ienouiinator ? [281
MULTIPMCATION.
tholonLy^""'^ * '■'•''°"»'' ly » ''hole number; or
ExAMPLE.—Multiply f by 5.
37. ReASOIV of the TfTTn? T 1 .
we are to add the multS^ca^ r«" T?^^oi*^ ^^^ ""'"ber.
as are indicated by the mu t?pSer?^but ILf?/' V^^^^y times
a common denominator we must add ftn ""^^ factions luaving
put the common denominatorTn'dfk^o'^rX^^^^^ ^
whTchcoSSe'tL^Scin'^o'^^^^^ "P^^^'«"'«f *hV integer
multipliertheir s"zeSTunoZZ"''n '^P^?f "^ ^^ *h«
be the same thing to incrlR7fh!?l^ • l^ ^^^^^ evidently
Without altering %l,eirS'^!^S,lT ', \" ^^"'^l «^t««t
dividing the denominator bvTh?l'^°"^'^ ^^ ^^^^t^d by
AX5 = . This will become Jll. ^'''° i^^^^^P"^^ 5 thus
the fractions resulting W Uf C£^t^o^otfr3^^^^^^^^
common denominatorfor ?? /=lX6\ ^^^ 4 •__ 4 ^T
will then be found equal ^^ ^ ^^ '^' '''' ^ (""15X5)
denornl7or'Srnu£ber'o?f*'P"^^ ^^ not contained in the
the method given in th?rf/;! """' expressed by an integer
, The rule wiirevUntly anSv""?/'.^' T"^ ap^plicable ^'''
Pl>ed by a fractions"fce^£ Ir^^ '""^T^ '' *« ^e multi
whatever order the factorrart'tn? rr«L'1"'^^i^ '^''^'^ i»
VULGAR FRACTIONS.
153
38. Tho integral quantity wlilch is to form oiio of
the factors may consist of more than ono deuumiuatiou
ExAMi'LE.— What is tho f of £5 2s. \)d. i
£ 8. d. k 5. d. £ s. d.
5 2 Ovj ^5 2 9x2 _3 g
0.
1. fX2=l.
2. 5x8=6^.
3. f,Xl2=:10J.
4. Jxl2=91.
5. VVx30=14.
11. i2x«G=34.
12. i«x20=l9.
13. 22x=4f.
14. AXI7=U
15. l43xH6i.
EXERCIBKS.
6. 27x1=12.
7. Axl8=3«.
8. 1X8=71.
9. 21xiJ=9.
10. 15x1=3.
16. Plow much is ^^ of 26 acres 2 roods .? Ans
20 acres 3 roods,
17. How much is \^ of 24 hours 30 minutes } Ans
7 hours.
18. How much is /jVa of 19 cwt., 3 qrs., 7 & .? ul7w
7 cwt.,3qrs.,2 1b.
19. How much is if of dC29 } Am. £\y ==£Q 195
39. To multiply one fraction by another —
Rule. — Multiply the numerators together, and under
their product place the product of the denominators.
Example.— Multiply  by .
4 5^4x5 20
9 6 9x6'^54
40. Reason of the Rule.— If, in the example <,nv..n, wo
were to multiply f by 5, the product (^^O would be tinea
too great— since it was by the siMh part of 5 (^), wo should
have multiplied. But the produ.;t will become wliat it ought
to be (that IS, G times smaller), if we multiply its denominator
by b, and thus cause the size of the parts to become 6 times less.
_ We have already illustrated this subject when explaiu
mg the nature of a compound fraction [11].
20.
21.
22.
23.
■LvS 3 5
l^Xt!"
^ ^ l^A. — '8'
I v^ .1
x^=
EXERCISES.
XV4 4 8
48 1
24. 1^X4!
25.JXfXfV=^5
ffS
27
3 14 vl?7.
3^3
si) ?.4
28.
29.
30.
31.
32. How much is the %■ of 3. > y} ,,? i
X V 1 1
fjXA=fV
I 2 ^ 8 — 1 6*
fins. i.
33. H ^w much is the f of f ? A //y
t ; '
■f^"'
IM
Vl/LOAR FRACTIONa.
of a fraotinn W "' "• . ^e8»"«8, the multiphcat oa
that of division ; and the number said in hi . uv^i
must bo made loss than boforr ^' multiplied
mkfd n^u"^^^ ' *'^^^'^"' ^^ ^ ^'^ ^er by a
raf/tn'T;?^*^"'^ '"''','^ °""^^«^« to improper fractions
r24J, and then proceed according to the last rule
Example 1.— Multiply J by 4.
"*!==*? J therefore ^ x 44=A y * > ~ 1 2 3
Example 2.— Multiply 5J bv 63 *'^"
52=V. and Gf=3^ tterefore 52x6f=:V x^^=^^^
„. EXERCISES.
37, Ax8Iy» v.. '■=;,.. £' Pixl3x6= SoTiJ.
^ 44. m. is t.o p.oa„et of e:'^1tt77>
^45.^ What i. tho product of f of f , and  of 3f »
8 4' #
r
f
. ULOAR FRACnONS.
105
44. If wo perceive the numerator of one fniction to
bo tho same as the denominator of the other, wo may,
to perform the multiplication, omit the number which
m commou. Thus f X5 = f.
«.;'!" 51* '* ?f, *""'® f 'livJding belh tl.e numerator ami fbM.o
mmator of the product by the saiuo nuiubuv— uud theicfore
does not alter its value; since lut^itioia
o^y=
5
ti~i}'
4.). Somctnnes, before performinju; the mnltiplic..tion,
we can reduce the numerator of one fnicti.>n nnd I ho
denominator of another to lower terms, by divi.Iin '
both by the Ranio number :— thus, to multiply i by •» ^
Dividing both r; and 4, by 4, we get in their placos,
d and 1 ; and the fractions then are A and J which
multiplied together, become ^X 4 = 7'. " '
tnJni^/" *^'^'''''P« as dividing the nuincrutor and denomina
tor of the product by the sanio number ; for
8^7bx7j42x'7 V=2^7/ ^H'
QUESTIONS.
1. How isafrnction multiplied by a whole number
or the contrary ? [36] .
2. Is it necessary that the intccror which constitnfos
one of the factors should consist of a single denomina
tion.? [38]. °
3. What is the rule for multiplying one fraction by
another ? [39] . "^
4. Explain how it is that the product of two proper
tractions IS less than either .? [41].
5. What is the rule for multiplying a fraction or
a mixed number by a mixed number ? [42].
6. How may fractions sometimes be reduced, before
they are multiplied ? [44 and 45] .
166
VULGAR FR ACTION'S.
WVlSIOxY.
tl.o whole 3^ and tui;';"'","' "'" «■''*""" h
inerator. ' I^"' ""= Product under it, „u.
Example. — ?i4 ^ ^
47. ReASOJV of THF TJirrr. Ti t .,
for instance, is to make ifSr '''"^''^^^ '"^ quantity hv 8
^t is evi.leut that if, wwl L i lo .?'^''"'^' *''"'^" ''^'^'"ro J3 t
same, we make theiTlhA If^'' \^'^ '"'"'^'''' ^'^ *''« P'lrts the
Jteelf 3 times Jes.s !Urct to muU! f ''..^^ T^^^^^ theSvact a
is to divide the fraction bv tL i,, ^'"^ * '^ denominator by 3
A similar cifect wT k^ f """; ^^umber. '^^ ^'
^iule we leave the .^. of^Sp^t" ;^ ^ "^^^f!. " if.
^.. 3 times less; thus '^iJ^/J "^ "'^'^^ ^^
numerator is not nh„ '^ ", '^ ~>' "^'"^ ^"'^° *^'«
of ^ comi.l,i lZfoT[l'2] " '^'^ ''^P'""'«i ">o nature
1. II_i_9 4
2. 4^^"7
4 ^9=^.
ESERCISK3.
«• H8=,,
12. ^,M4_/''^^
7. A>14= • .
wlieu we multiply rdtido?s'° ^'^ ? "^^^^or, th.,,f,
natorbj the sa no r mm or w ?""^*^^'^l«^ ^"d denonu'
«nce we then a tlTZl r ""^ ^^^'^■^" ^^^^ ^^^1"«
dcoroase it. ' '" '"'"" *^^^> ^^I^^^^J increase and
4f' To divide a fraction hy a fraction
llur,E.— Invert the dlvf^ri /"''^'^^^n—
'^o^ed), and then placed ^i/tL7^r ^' '^ ^^ ^^■■
multiplied. ^ " ^"^ fractious were to bo
VULGAR FRACTIONS.
157
Example. — Divide  by f .
5_^3_5 4_5x4_20
7*4 7^3 7x3~2r
Reason of the Rule. —If, for instance, in the
just given, we divide  by 3 (the numerator of the
we use a quantity 4 times too great, since it is not I
the fourth part of 3 () we are to divide, and the
(£y) is 4 times too small. — It is, however, made what
to be, if we multiply its numerator by 4 — when it
1^, which was the result obtained by the rule.
50. Tho division of one fraction by another may bo
illustrated as follows —
example
divisor),
y 3, but
quotient
it ought
becomes
5
.
3
'.A
«n
•a
*
ns

—
The quotient of frf must be some quantity, whicb,
taken threefourth times (that is, multiplied by ^), will
be equal to f of unity. For since the quotient multiplied
by the divisor ought to be equal to the dividend [Sec ,
II. 79] , f is f of the quotient. Hence, if we divide tho
fivesevenths of unity into three equal parts, each of
these will be owefourth of the quotient — that is, precisely
what the dividend wants to make it fourfourths of the
quotient, or the quotient itself.
51. When we divide one proper fraction by another,
the quotient is greater than the dividend. Nevertheless
such division is a species of subtraction. For the quo
tient expresses how often the divisor can be taken from
the dividend; but were the fraction to be divided by
unity, the dividend itself would express how often the
divisor could be taken from it ; when, therefore, tho
divisor is less than unity, the number of times it can bo
taken from tho dividend must be expressed by a quantity
greaUr than the dividend [Sec. II. 78] . Besides, divid
ing one fraction by another supposes tlie multiplication
of the dividend by one number and the division of it by
another — hut when the multiplication is by .a ffrea';er
4
m
'hfl
fW
•4 2
158
VULOAU FRACTIONS.
must beicrea™!, " ''""""'^ ""^ '» •"> iviJod
13. ^^3=I^!
14. 4^2=1
15. 141=13.
EXEnClHKS.
?7 l"^H' I i»fi^=i^'
18. j#i=ii. 21. ?j.jrr
62. To divide a whole number by a fVaotion'
minator of the product '" ''' '''''''''''''' '^^^ ^eno
Example.— Divide 5 by ^
5^3^5x7^a5
' 7 o "" "5" •
This rule is a consequence of Hm i..of . ^
tcr may be considered a" "fr t\ V •'' '^"'^ ^'"^^^ """i"
iDinator [14]; hence sl" J'^n^'x "'^ """''^^ ^'"' '^'""
b^'<S; dJn^SS U!'' '^''^'^ '^' «'11 consist of
Example. Divide 17.s'. ShJ. }>y 3
22. 341=6?.
23. ll^/*.=,i.'u
24. 42;;j,=gG4.
I\')SRCISES.
25. 5ii=,')i
31. Dmde £7 IGs. 2d. by a yi., o,. , / ^ , 7
32. Divide ^8 13.. 4d by r J ' f n J"^^'^
3.Prv^e^5 0..1..by^r j^;; ^^^ ^
o^:S. 10 divide a mixed nmnbnr h,r o T.ri i ,
or a fraction— '^•^ ^ ^'^^^^^ number
and 49]. ■'' ^''''^'' '' ^^^"^^^ directed [40
Example 1.— Divide 9f by 3.
9?^3=9f3f■;^ ^3=341=3'
Example 2.— Divide 14^3 by 7 ^' '''
'.4''='*^'' therolore lM^2=',FM = '^X^='?«
'■■.■^n:'^:'^0^m^f ''■:■'''
.5
.'J
VriLrjAR FU ACTIONS.
159
54 llEx^oN OF THK RuLE,~lji the first cxnniplc we have
divided each part of the dividend by the divisor auu added
the vesultswhich [Sec. II. 77] h the same as dividing the
^•liole dividend by the divisor.
In the second example we have put the mixed number into
a more convenient form, without altering its value
EXERCrSES.
34. 8^17=f.
35. 51^.3=lK\
'151
rs
.1^ fl450
■''Tf.
39 433S_:_41 ,
40. iifi^MSffir
41, 188X^^5— 19*773
42 loVJ^irliim,
43. 18±Ail=ii«7 •'"^
'■v^'
55. To divide an integer by a mixed number
Rule.— Reduce the mixed number to an improper
fraction [24] ; and then proceed as already directed
1 52].
Example. — Divide 8 by 4§.
41=%^ therefore 8f.4f=8^ 2^3=8 x./j^Uf.
Reason OF THE RuLE.It is evident that the" improper
fraction which is equal to the divisor, is contained in the divi
ilend the same number of times as the divisor itself.
' 1
44.
45.
4S.
49.
46 14^l=7^V.
47 21^ll3*=iii
. 3
5^
n^d.
EXERCISES,
5f.3l=lf,
16i 1112—113 3
Divide £7 16s. Id. by 3i. Ans. £2 6s.
Divide £3 3s. 3d. by 4i, Ans. Us. O^d.
56. To divide a fraction, or a mixed number, by a
mixed number — "^
ron'''''''75'''^''''' "''''',^ numbers to improper fractions
LJ4J ; and then proceed as already directed [49].
Example 1.— Divide  by 5 J.
6l=.f, therefore  ^5^=^^5/X,3><^^^,^^
Example 2.— Divide 8j9j by 7.
8A=fJ, and 7=V, therefore 8X^7#r^"_..yr.7 w
47 Reason of the Rule. — We (n.<i in tim loof •, i \ ,
ehango tho K,ixed nvmbcr, into *o h e" fm J ' cot S'j
dmcledwithout, however, altering their value °'"'"'"'"'"''
o 11
fl 56 2
'1
i
^ij
160
VCLGAR FKACT/ONS.
50. ~'^.TJ. — •lii
51. ^1441=11*
52. J<.^3A=Vi3
53. U^jlll^¥^^
54. MJsCfT
2 • ^3 ^32.
EXERCISES.
55. 82 rV— 26 /'=<= '?«■■■
50 ioM\i^*'',ti<'>^^
57. ^^:^8^=::i^3»^^^"'"
59. 2J.34+i^e..h,
58 When the divisor, dividend' o/hnfJ. "
poand, or complex fraetimis ^ ^'^^'' ^'"'^ "^""
Wrsr^'p^Xm^'ft^ del,, to sin,.e
which are cLCuTL^d tt 7^ ^« ^^'"'^^^
are complex ; tLntr'ocTe'd ^^l^^S^Z^^:^)
^Example l._Divide 4 of  by f . '" ' ^"''^
f of «=3o {39j^ ^j^^^^^^^^^ fX«lo=^n_no^._,,,
Example 2.— Divide ~ bv •'
i=4% M, therefore g=^^.=^x=^,
o
rn ^ o EXERCISES.
01. 4fil— S^VJ* «;n43
62.
'8~ IT *
21
2 •>
63. H^sy_7_n7
97 • 3'^i5— im
o4. . — 1 S.~ron
UO. — _i;.2'v « OfTt
19 '"
<i
60. ~_i.2v5 Q231
7
QUESTIONS.
I. How is a fraction dived bv an integer ? r4fil
o. J^ixplam how it ocputq f^.;* *i """.• L'**J.
fractionsis sometimes greZthal t fT' ?i '^"
4. How ;.! , »i,,i 8'"'"" man the divider,. J fsi i
[62]. ' " "'""^ """'''«'• divided by a finetlW?
ggj' number, by a mi^d number ? [55 and
to.I r'l;tVot oSpTe;t^^^^^^^^^^^ [fir. "
.*^Jr:..
.'■^ „ :t^ .=
VULGAR FKACTIONS.
16J
MISCELLANEOUS EXERCISES IN VULGAR FRACTIONS.
acres, 3 roods ?*
Ans.
1. How much is ^ of 1S6
20 acres, 3 roods.
2. How much is ^ of 15 hours, 45 minutes ?
7 hours.
3. How much ir, //f% of 19 cwt., 3 qrs., 7 Bb .? ^7w.
7 cwt., 3 '^rs., 2 tb.
4. How much is ^V/^ of £100 ? Ans. £3Q 95.
5. If one fanii contains 20 acres, 3 roods, and
another 26 acres, 2 roods, what fraction of the former
is the latter > Ans. ^^\.
6. "What is the simplest form of a fraction express
ing the comparative magnitude of two vessels — the one
containing 4 tuns, 3 hhds., and the other 5 tuns, 2
lihds. ? Ans ' "
JlS.
o n
1. What is the sum of  of a pound, and f of
a
shilling ? Ans. 13^. lOfri.
8. What is the sum of 5. and ^ul. ?
9. What is the sum of dCi, a/
36'. IfirZ.
10. Suppose I have ^ of a ship, and that I buy y\
Ans, 7j\d.
and y'jf/. } Ans
more ; what is my entire share ?
Ans. Ji.
11. A boy divided his marbles in the following manner :
he gave to A ^ of thorn, to li j\, to C i, and to D i,
keeping the rest to himself; how much did he give
away, <and how much did ho keep ? Am. He gave away
tVo of them, and kept j\?^.
12. What is the sum of  of a yard, j of a foot, an(J
4 of an imh .' Aiis. 7 inches.
■'3. Wha. is the difference between the  of a pound
and o}d. ? Ans. lis. 6^d.
14. If an acre of potatoes yield about 82 barrels of
20 stone each, and an acre of wheat 4 quarters of 460
lb — but the wheat gives three times as much nourish
ment as the potatoes ; what will express the subsistence
given by each, in terms of the other ? Av:<. The pota
toes will give 41 } times as much as tlie wliL^a, :, and tho
wheat the ,£j\ part of wliat is given by the poiutoes.
15. In Fahrenheit's thoniiDnioter "there arc 'SO de
grees between tho boiUng and freezing points , in that
163
iniClMAL rUACTIOiVS.
of Rcaumav only 80 ; what fraction of a do.rcc in tho
lat or oppresses a cjegrcc of the K,r,ner r X? ' " *^''
isak.'nt ia ^T^' ^'^ 'f '''''' ^" t^^« United" Kingdom
w about 34 inches jn aepth durino tlio voir in fl,o t ,•
of hi boii /!'" """"":>"'''. "■' 22,480 ; what fnlctiou
1. «i^ "* ^ '"■S"""! ''^l"™^"^ tl'^'t of Chimbora.o ?
jjoi, w^i r » r zr To=x™rri
fraction of the latter ? ^w^. f.i. "^ '^prc.ssca as a
DECIMAL FlliVCTlONS.
59. A deun.:il fraction, as aheady rcmarlied Tl^l
Sinoo the division of\Su™l;Cof\t ci:!! tc'l™;
by .ts d,3n„™.„to,fro,„ the ve..y natu,ror ,!^ ^ S
infnf fl J""'^ Praformod by moving the do.nmil
60 It is as inaccurate to confound a decimal fraction
with the corresponding decimal, as to confoZd a 4w
fraction with its quotient.— For if 75 i< th. fi ,
from eit,:^"""™* °' ^ "'• "' ^^ "'l Ciually'distior;
mnl 'f. ^."<"^™»I « cliangcd into its correspoudi,,.. deci
ml fraction by p„tting unity witl. as manv cypfcs 4
f»"^ .';'''"" 'r^' "■'' '!=«'"■■>■: point. Thus Oe^fi^
I O 1) ) '^^
,•5 (1 4 r,
I J
DECIMAL FnACTI0N3.
163
gi'Go in tlio
I Kingdoiu
the plains ;
mt fractioa
Iiigh, and
at fractiou
limborazo ?
fissure or
bet as tlie
lountairis ;
sscd as a
:ed [13],
lilt hand.
fra<.ition.
I fraction
nutation
deoiuial
ho equi
facility,
ty by a
nt three
fraction
a vulgar
q.wtknt
; so also
distinct
ng deci
)hers as
nator — •
5646=
^2. Decimal fractions follow esactif UiC same rules
as vulgar fractions.— It is, however,* generally nioro
convenient to obtain their quotients [oG]^ and then per
form on them the required processes of addition, &c.,
by the methods already described [Sec. II. 11, &c.]
63. To reduce a vulgar fraction to a decimal, or to a
dediiial fr actio, i —
IluLE. — Divide the numerator by the denominator
tins will give the required decimal ; the latter may be
changed into its corresponding decimal fraction— as
already iokicribed [61].
ExAMPLK 1.— Reduce I to a decimal fraction.
4)3
Example 2.— What decimal of a pound is lid. *
7^/.= [17] £i^ but £5Vo=^C0032, &c.
This rule requires no explanation.
EXERCISES.
1 7 _8 7_5
3. ^V— 36.
4 i' — jK'L
^' 4 100'
5. •!
625.
6. ^^=973&o.
7. J=5.
9. j«,/.,=90476, &c.
10. =.8.
11. /^=5625.
13. Ileducc \2s. Gd. to the decimal of a pound. Ans
625.
14. Reduce Ids. to tlie decimal of a pound. Ans. 75
15. lleduce 3 quarters, 2 nails, to the decimal of a
yard. Alls. '875.
16. Ileduce 3 cwt., 1 qr., 7 lbs, to the decimal of a
ton. Ans. 165625
64. To reduce a decimal to a lower denomination .
IluLE. — Ileducc it by the rule already given [Sec.
III. 3] for the reduction of integers.
I^lsAMPj.K 1.— Exjjress £6237 in terms of a shilling
•6237
20
. li
ft
'in
11
Answer, i2'4740 shillings=.CG237
B! 'if
HI.
164
DECIMALS.
Example 2.Rcducc i;.9734 to shilling. &c
•9734
20
194G8() 8hIllings=X9734.
5G160 ponce=4C85.
4
24040 favthinffs=G10^Z
Answer, X9734=las. 5 J,/
ro^SoI'^^^^^^^^^^^^ a, wc. given
of » shilling by rrroJuccs rt ■^'"'''l,''^ "8,'l'° <1««"""1
J'Wmy. MultiDlvinff t^ryj!, ? '^.t"™ ""'' "'" ■)«"»"'>' of a
EXERCISEa
23. WJuat is tlie value of £80875 ? yl,,,. 17,. 4^,1
4. What IS the value of £d375 ?
20. I ow much is 875 of a yard. ^ ^M,s^L^^^^^^^^
27. AVhat is the value of £05 ? Ans. Is
^^f^ How much is 9375 of a cwt. ? Ans. 3 qrs.,
29. What is the value of £95 } An^. \Qs.
30. How much is 95 of an oz. Troy ? Ans 19 dwt
31. How much is 875 of a gallon > Ans 7 nints
28'%'':^.r''' '''''' ^''^ ''^y^ ^1// hours,
;J^' !^!'^ f^"owing will bo found useful, and—befnr,
n nnatcly connected with the doctrine of SctionT^
may be advantageously introduced here : '^''^^^^"^
io imd at once what decimal of a pound is enn.V.
Whor;l"""'^" «f ^^^^. ponce, i?r '^"""
\V on there is an even number of shillings '
a pound" "''^'' '^''''' '' ^' ^'''^^ ''' ^^^y t^^^tbs of
DECIMALS.
165
ivcre given
i it to shil
l»o decimal
cinial of a
I'educcs it
175. 41(1
■ , 2 nails.
• 3 cwt.,
3 qrs.,
19 dwt.
pints,
hours,
ins. 1 5
— being
tions — •
cqniva
?tlis of
E.VAMrLE. — lC)S.=£8.
Every two shillings areciiual to one tenth of a pound; there*
fore 8 times 2s. are equal to 8 teuths.
67. Wbon tlio nunil)er of shillings is odd —
lIuLK. — Considor half tho next lower oven number,
as so many tenths of a pound, and with these set down
5 hundredths.
Example. — 15.s'.=£75.
For, 15.? — 14S.+1.S. ; but by the last rule 14s.=cC7 ; and
Binco 2s.=l tenth— or, as ia evident, 10 hundredths of a
pound — l.s.=5 humlredths.
68. When there are pence and farthings—
K,uLE. — If, when reduced to fjirthings, they exceed
24, add 1 to the number, and put the sum in tho second
and third decimal places. After taking 25 from the
number of farthings, divide the remainder by 3, and put
the nearest quantity to the true quotient, in the fourth
dechnal place.
If, when reduced to farthings, thoy are less than 25,
set down the number in the third, or in the second and
third decimal places ; and put what is nearest to one
third of them in the fourth.
Example 1. — What decimal of a pound is equal to 8,J(L T
8J=35 farthings. Since 35 contains 25, wo add one tc ■
the number of farthings, which makes it 30— we put 36 in
the second and third decimal places. The number nearest
to the third of 10 (3525 farthings) is 3— we pvit 3 in the
fourth decimal place. Therefore, 82=£03G3.
Example 2. — What decimal of a pound is equal to 1:^(?. 1
1^=7 farthings ; and the nearest number to the thir*! of
t is 2. Therefore ltZ.=£0072.
Example 3. — What decimal of a pound is equal to 51(1. "?
5](/.=21 farthings; and the third of 21 is 7. Therefore
 3](i.=i:0217.
69 Rkason of the RuL,E.We consider 10 farthings as
the one hundredth, and one farthing as the one thousandth of
a pound — because a pound consists of nearly one thousand
farthings. This, however, in 1000 farthings (takeu as so
many thousaudtlis of a pound) leads to a mistake of about 40 —
Binco ,fil=(not 1000, but) 1000—40 farthings. Hence, to a
tliousaud favihings (considcveil as thousandtliB o^ a pound),
ii
/ /THs: .\i
* '4
1C6
11 at'
CIRCULiTlJtG UE^.jiAl.,.
correction "liouhhtill bo I,, , M ,";;•■ "• " "'''"'"
or .«.«w one, i„ t,,o/„°!;?raii;f„,'fr;s;'' ''*
KXERCISEfl
18 7 i ft. ==:£ 0822.
J9. ^'7 5*. I0rf.=je272916
tljo •ne
tiiat, ua
must be
nunibor,
that the
number,
20. 4,,.3,/.=£.7i55
^i. £42 ll5.6R=i;.42677
^, in an/dtilrora'X^i:::!^" "^ *"'»o. ponce,
tJonsMer the digit in ttc™?! ?,,'«" '',""'' °™
6, If It 18 not loss thau 5^ r,„„i /f''''' ^n'>'™o'ing
;«.unito of farthings ; and s, b ta 1 "" •, f' '" ""^ "'"^
>f It exceeds 25. *"'" "'"'^ f™ni the result
ExAMrLi:.je6874=I3s 9rf
4r^^^t^::^^^:^i^^ « widths
tiie remainder (reduced fn f i, ^^"n^/redtJis and adding
sandths, we have^T thousandtt r^^^^^^l^ *^ *he tS
exceed 25, we subtract Svtfe i™ ^vhichsince they
of farthings. ^6874 theJefoS i T' ?^/^ *^« ""™be?
tarthmgsor 136. 9d. '"^^®^"^®' i« ^qual to 13^. and 3G
" ^^ ^^"owa fro. t.e I.t three^being the reverse of
;:i
CmzVLATim DECIMALS
''w»yi;ra"f 'c "ottro?^ ?^*'?'' [See. H. 72],
number by a„otlor?ll ST' ".''° '""''^ ™«
'ta..Iy recur, 0^0^,^ a'^^^E' T *'^'"'; ™"
CIRCIJI,ATI.V(; BE(,...IAL8.
Ul
docimal is i)roclue( il.— Tho decimal ia euiJ to bo termi
nal, ,'.' there is an exact iuouciit — or oue which loaves
no romaiiider.
72. An iutormiiiato dcciiiKil, in which only a ingle
figure is repeated, is callc ' a rppettnd; if two or more
^ligits constantly rccnr, thoy tonn a. periodiai/ '' \,
Tiins ;77, &c., is a repcfnnd ; hut 59759/, ..o. i.s a
periodical.^ Vov the siko ot brevity, tlie repeated digit,
or period is set down but once, and may he marked as
follows, 5' ( = '555, &c.) or M'jri' ( 193493493, &.c.)
Tlio ordinary method of marking i\n: period is sonie
wliat different — what is liore given, howcv<T, seems
preferable, and can scarcely be mistaken, even by those
in tlio habit of ^ ^ing the other.
When the d imal contains only an vnjivile. Mt —
that is, only tlu; repeated digit, or peiiod — it is u pure
repctcnd, or a p?/. rc_ periodical. ' But when there is hoth
a finite and an infinite part, it i.s a mixed repetend or
viixe.d circulate. Thus
j*'^ (=V)oo, &c.) is a pure rcpotcnd.
'578' (=57iS88, &c.) is a niixod ropotond.
'397' (= o97397;">97, ka.) is a pure circulate.
8G5^G427r(='8G5G427164271G4271,&c)is a mixed circulate
73. The number of digits in a period must always ba
less than the divisor. For, different digits in the perioci
suppose different remainders during the division ; but
the number of remaind;MS can never exceed — nor even
be equal to the divisor. Thus, let the latter be seven : the
only remainders possible are 1, 2, 3, 4, 5, and 6 ; any
other than one of these would contain the divisor at
least once — which would indicate [Sec. 11. 71] that the
quotient figure is not sufficiently large.
74. It is sometimes useful to change a decimal into
its equivalent vulgar fraction — as, for'instancc, when in
adding, &c., those which circuhite, we desire to obtain
an exact result. For this purpose —
IluLE — T. If the decimal is a pure rcpclevd, put the
repeated digit for numerator, and 9 for dcnominato..
II. If it is a 2>ii''& puriodimly put the period for
numerator, and so many nines as there are di<;its in the
period, tor denominator.
H
IMAGE EVALUATION
TEST TARGET (MT3)
i:
■V'J
•ip M^
^ A'
W
W^r
/
^
1.0
1^
1^
m
\m
142
1^
136
22
\L.
liiH
I.I
1^ RRffil
1.25
12.0
1.8
14. ill 1.6
V]
<^
%
/a
/a
w
om
9
V//
Phntnoranhir
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 8724503
V
iV
^
^^
\
\
A
6^
>^
m
'' '  ,""■ > »'*
S"y ■ .
„";S,' >. iC'""
IG8
CIRCULATING DECIMALS.
^^ Example l.What vulgar fraction is equivalent to 2' 1
Example 2. — What
*7S54'? Am.
7S54
VW99
vulgar
fraction is equivalent to
we multiply two equal o7anHL?'""i?^^^^' '^*') ^^r i#
quantities'ihep?oclltwiarbceoVar "^"^' °^ '^ ^^^^
Fo?1:E ^i;;^ ^£^X^^T'' '']'  quotient.
OS 100 kundredhi3'\Z}ihTV^.^i'c,'^'^''^^ ^e considered
100 ten #/.«W "jM. Sefore t wi ^ "^^ ^' °^^^'«
will be one /.n S.aS/L ^ Z.^? ' ".T^ "'^ '^'^«"^»<^
remainder, must, in the /amP L ^ k ' ^"P *^" thousandth, the
eth.s; and the nexrquotiTt wIlL "'^^^'f P^^d as ten milHon.
and so on wi irtl?e oS ni r .""^ '"^'^''''*^'^''«^ OOOOOI
will be Oli^mi^Som^^^ together,
by ^01'. ^ ^^wi^^vc, or 010101, &c.representod
■ 2^5 (==37XB'fl=37X.^0r) will eive 337^7 Xr^
quotient. Thus ^■o/o/, &c.— or
010101, &c
37
a 3
70707
30303
Tr, +1 .. S73737, &c.=37vm'
digits as a period, will bl eauaVto a v ,l ? ^?"°« ""^ '«
603
3003003003
600600(5006
5005005005
T« *!,« ^6^5^3503503, &c=5fi3v\nni
b >e a cucuiating decimal having these
CIRCLI.ATING DECIMALS.
169
J'pita as a perio(J. — And, consequently, a circulating decimal
linving any three digits as period will be equal to a vulgar
truction having the same digits for numerator, and 3 nines
lor denominator.
We might, in a similar way, show that any number of digits
divided by an equal number of nines must give a circulate,
«ach period of which would consist of those digits. — And,
Consequently, a circulate whose periods would consist of any
digits must be equal to a vulgar fraction having one of : s
j'.eriods for numerator, and a number of nines equal to the
number of digits in the period, for denominator.
76. IF tlic decimal is a mixed repetond or a mixed
nrculate —
lluLE. — Subtract the finite part from the whole, and
set down the difference for numerator ; put for deno
minator 80 many cyphers as there are digits in the Jinife
part, and to the left of the cyphers so many nines as
tliere are digits in the iiijiniie. part.
ExAfdPLE. — What is \i\Q vulgar fraction equivalent to
•97^8734' ?
There are 2 digits in 97, the finite part, and 4 in 8734,
the intiaite part. Therefore
97878497 978G37 . ,^ • , , ....
~mm~=WMO' '' *^^ '''^1'"''''^ ^^'^Sar fraction.
77. Reason of the Rule. — If, for example, we multijdy
•97^8734' by 100, the product is 97 •8734=97 48734. Tliis (by
the last rule) is equal to 97o^' which (us Ave multiplied by
TOO) is one hundred times greater than the original quantity —
but if we divide it by 100 we obtain TVo+iTsMfis'fn. "wlucli is
equal tjio original quantity. To perform the addition of y^,
^^^'^ irf sffiW' ^e must [19 and 22] reduce them to a common
denomin?».tor — when they become
97X991*^00, 878400 97X9999, 8734
99990000 ' 99990000"" 999900
97x100001 , 8734
100001)
970000—97
' 999900
97x1000097
8784
"999900"^
999900 ^999900 ~ 999900
8734 97873497 978637 .
999900 "^999900~ 899900 ""999900' "^^^" ^^ exactly the
result obtained by the rule. The same reasoning would hold
with any other example.
EXERCISES.
7.
1.
2.
3.
4.
5.
6.
•^8'=5L
• 73'=T^
_14 5
^057'=rg?X
^145'=
^057'=
8.
9.
10.
11.
12.
./)74' 574
uu .u — onno
•147^658'=:iil^ii
87549vG5'=875*^^V'
3018275G'=:301^;^^a.
170
^•^KCULATINO DECIMAL,.
vaient vulgar fractio, '^d "° *''"'"''^ "> "««• o„"i.
''«'«, &0, liie other <le",l!" '""■^ ?''''' "id subtract
put Joivn so jaanv of *i "'"'«'/ takin ca.rtl
acourac,. """^ »' 'leu, as will ^secure "suffieio,;:
dered as :^2i£2«;s\ .„ *• ^tus a (const
5 >'«'^^g™o.act,„otiont;soalso
W'U i (considered as ™i£2*?x ,
2 ) i^ni \ will not siyo
»»; fe4 (considered as'^MSSl^, „, IMJmndn^itl.s
^ "■■.•'^nkr reason 4 «/,„
'"i>oo4 (considered asl°ii!s;. 400hXdS:
%l ''rj„1(';' ^■''.'"Ij contain 7^ ^^ ''
»« will be eqtl to™e Zlt™ '""^y depi,nal places
eontamed as factor i„^S 1' "™?''°' "f twos, or fives
therefore i2j£f^ " ^^ (^XS; ; and
'^^ place 'see. ^7^1^^""" ''^' ^"^ '''
quoaeoi. ^^ ^^J), that n, ,,,, ^.^.^^j ^^
> 2X2/ ^^" gJve two decimal places • ho.
CIRCULATING DECIMALS.
171
'■gJ, it is not
tlieir ofjui
nd subtract
"ig cava to
■e sufficient
?ar fraction
' its Jowost
ctors (fac '
ictors, can
or noitlior
tliese — as
oir inulti
ictly con
I (eonsi
* ', so also
' noi givo
idredths.
ct quo
ii'Gdtlis,
places
1* fives,
riginal
'found
; and
in the
al as
5auso
■itor,
•s so
30 tenths
niany tenths ; for — ^ (=4 ) cannot give an exact
quotient — 30 being equal to 3X2X5, which contains 2,
but not 2X2. It will, however, be sufficient to reduce
, , , , 300 hundredths
the numerator to hundredths : because .
' 4
will give an exact quotient — for 300 is equal to 3 X 2 X
2X^X5, and consequently contains 2X2. But 300
hundredths divided by an integer will give hmulredths —
or two decimals as quotient. Hence, when there are two
twos found as factors in the denominator of the vulgar
fraction, there are also lioo decimal places in the quotient.
4*V \r^ o \^ o sy o sy r j contains 2 repeated three times
^ ,«X'*X'^X<J
as a factor, in its denominator, and will give three
decimal places. For though ]0 tenths — and therefore
6X10 tenths — contains 5, one of the factors of 40, \\
does not contain 2X2X2, the othr;> ; consequentlj'
it will not give an exact quotient. — Nor, for the same
reason, will 6X100 hundredths. 6X1000 thousandths)
6 X 1000 thousandths
will give one — that is, j^ (=4V) ^'"^
leave no remainder ; for 6 X 1000 (=6 X2X2X2X5X
5X5) contains 2X2X2X5. But 6X1 000 Ihausandths
divided by an integer will give thousandlhs — or threa
decimals as quotient. Hence, when there are three twos
found as factors in the denominator of the vulgar frac
tion, there are also three decimal places in the quotient.
81. Were the Jives to constitute the larger number of
factors — as, for instance, in /^ jf ^j, &c., the same reason
ing would show that the number of decimal places would
be equal to the number of fives.
It might also be proved, in the same way, that were
the greatest number of twos or fives, in the denominator
of the vulgar fraction, any otlber than one of those num
bers given above, there would be an eqaal number of
decimal places in the quotient.
82. A pure circulate will have so many digits in its
period as will be e(pial to the least number of nines, which
would represent a quantity measured by the donoraina
172
CIRCULATING DECiaiALS.
bo equal to a fraction ),,„/„ J """ '"<"' " e'reulate will
that IS, It will be eaual +n =. ^ • ^^^ ^^enomiuator—
of which (the ierlToff}T' ^T'T^ '^'^ numerator
ii^ the numemtor of the ^,''''"^' '^ ^^^ ^' «^ ^'^^J
quantity represen ^d VlL^ST^^r^^^^?^^^ ^ ^^^
l^r if a fraction having a^i^'nl '^ '*' ^^"^""nator.
another which has a alpf f ^ t""'"""'"^^^ ^« ^''l^^l to
of the latter is to the slm^'. '" ^'T''' ^^'' numerator
the former^in which ciTtho""?" '?'' '^^'" *^^^* ^^
nierator counteract ttrefe^^^^^^^^ ''^'^ the nu
denominator. Thus AVf , *^'^ increased sue of tlio
«f H is 5 times Teaterd^ ' i'TT^ " ^^^' numoratur
«nd 384615'=5 8 4 el?"! J ^V ^^"^e /^=.v3846ir/ •
and, thereforp Jk" V^ '^' ' A' '^^^°' ^^ ^^nai to If a# i5 .
_ u, tuertiore, whatever mult nIp '^ft.iRi • »»y*nf'" > 
J:8 the same of 13 —But qq S • ^f^^'' '^ ^^ ^> ^^''^'^^^
13, consisting of nin^s T? . ' f /^^' ^'''' "^"^^^P^^^ *'
Then take f?r numelato" sn .1 ' "' T'? "'^^^ ^' ^^^««
lesser number of ^ n'; Vs ^^ , i^ '"'"^y' ^^ '^^ ^« that
number of nines for its deomulltr'Tl.^'' that lesser
this new fraction will fir^l ^'^l'^'' ^.ho numerator of
equal to the origina fia^i^on "\ ^ ^''''?^ '^ ^ ^^''^''^^'^tc
different from 3846 5 Tf '^ ^"^ ^^t^"^ ^«^ Period k
circulate; there TrCefe'tr^r^^"' *^? ^'^"'
equal to r^^that is two rii!' T t^'"'"* circulates
for the same frac ioniwh.V f "'''' ''^^''^'^ ^' 'otiont,
is absurd to suppo eVa^^a^.v /' '""^'f^^' ^ 'nee it
multiple of 13. ^ "^ ^"^ ^"»^^«^ of nines is a
rarTwh'a^ nSf 2^t1"'fo'"^ ^^^ ?^^^^ '^^ ^te'
of the vulgar fraction^oduced tot ? '^! ^^^^^ominator
For f7fil n «».•* ^^uucea to its lowest terms
hand"i/'ti; 't??n' r\' "''• "^P''^ *»" ■■%>"
fition,oI,tainXmL 'S,.e:?r,rt; "' f''" "''^'"•
suppose the (ienominator of t„ ^^»' ';.VP'""s would
tain two, 0,. fivci^i:^ t sniiSrt;::
CIRCULATING DKCIMALS.
173
to its Icwost
3irculate wiJl
01' its nume
iiomiuator —
e numerator
be as maivy
tion, as the
snouiiuator.
is e(ual to
■ numerator
'an that of
of the nu.
' «ize of tlio
numcratur
momiaator
2.
'3846 If/ ;
0.4. 0L5 .
a u ? ft ft ) —
^, U9!i't>J.4
lultiple of
'r bo loss.
5, as that
i«it les'ser
^orator of
circulate
period is
culate of
Le former
•irculaios
'"'otients
■incc it
nes is a
a finite
>minator
IS.
he riojht
vulgar
'S would
tion to
facturn
could give cypliers in thnir multiple
of the vulgar fraction obtained from the
If there is a finite
the dimominat\7r
ciiculate.
84. If there is a finite part in the decimal, it will
contain as many digits as there are units in the greatest
number of twos or fives found in the denominator of tho
original vulgar fraction, reduced to its lowest term.s.
' Let the original fraction be /g. Since 56.=J2X2X
2X7, the equivalent fraction must have as many nines as
^vill just contain the 7 (cyphers would not muse a number
of nines to be a multiple of 7), multiplied by as many
tens as form a ])roduct which will just contain the twos a8
factors. But we have seen [80] that one ten (which adds
one cypher to the nines) contains one two^ or five ; that
the product of two tens (which add two cyphers to the
nines), contains the product of two twos oi fives ; that
the pror'.uct of three tens (which add three cyphers to tho
nines), contains the product of three twos or fives, &c.
That is, there will be so many cyphers in the denomi
nator as will bo equal to the greatest number of twos or
fives, found among the factors in the denominator of tho
original vulgar fraction.
]3ut as the digits of the finite part of the decimal add
an equal number of cyphers to the denominator of the
new vulgar fraction [7GJ, the cyphers in the denominator,
on the other hand, evidently suppose an e(ual number of
places in the finite part of a circulate : — there will there
fore be in the finite part of a circulate so many digits
as will be equal to the greatest number of twos or fives
found among the factors in the denominator of a vulgar
fraction containing, also, other factors than 2 or 5.
85. It follows from what has been said, that there is no
number which is not exactly contained in some quantity
expressed by one or more nines, or b;7 one or more nines
followed by cyphers, or by unity followed by cyphers.
Contractions in MtrLTiPLicATiON and Division
(derived from the properties of fractions.)
86. To multiply any number by 5 —
IIulb:. — Remove it one place to the left hand, and
divide the result by 2
174
<'"\ri:.\'TroN'<<.
KxAMrr.r;, . 7:!Ox/5~."^^h,^;..;^,,
^' 1<» imiltipiv \,y 25 _ 3
divide liM^'""'"' "" 'i'"^»t^'y two places to tho left, and
M\.\.AnM,K.— 0732x25.^ ''f.iaon..,, .«.,^n
^— , ; flH^rcfurc G/.!i»xi:5=G732x'""
*^ 'i'o multiply l,y 125 ._ " *
di^:'^';::::;:i;'i ^^^^^^^ ^j pi^^ec. to the loa. and
liiOAsox. — 125— ''^»" t) r Tr^r^^^^
KVAMl'LE G85 XTSsTT*"' 5"" ^T .,n . 
lOOx,/. * I'JUX, : theififore 085x75 = C85x
'H). To imiltiply by 35—
left STi!; l^^'t'S^tfo""?"! ^"^ I'^ *« tho
I'luce to the left. ^ ' ^ *''' "Hiltiphcand removed one
'f~fi'0. ' therefore G / 80Gx 35 =67806 x
p!; J'' t'^' '7'^"^ <^'^'^'>^ the multipliers
Fv^;7 v^ ^ '^^ '"' "^'"^'^^^"^ ^'^^tion. inverted
^L^.,...4.v.de 847 ,^ 5. 847..5=847^ V^47x
easy to divide, a. ro v^ultnAl'C"'^j',r^^". '^ ^'."^'^ ^^
liUAed number. " ^ "^ ^ •* r ^^> i^« crpnvalene
DECIMALS.
m
^=«^v"==:5'Vfto.
tlio left, and
o V ' " "
QUESTIONS FOR THE PUIMf,
1 Show tliat a decimal fraclion, and the
spond
^ X '
the loft, and
'SG5='7o.
"•f^ loft, then
hy 1.
5.
C75 = C85x
«cns to tlio
Jinovcd one
== 109740Q
= 67896 X
ornsclves to
ertod.
y'=847x
■ide hy the
1 wJieiii we
3h'er v'ill
is not so
riuivalciit
Iccmif
*"6 "'•'^"""'i «»>iu iiuu identical L^jyi.
2. How is a decimal chann;cd into a decimal frac
iion? [GIJ.
3. Are the methods of adding, &c., vulgar and deci
*nal fractions different ? [62].
4. How is a vulgar reduced to a decimal fraction ?
[63].
5. How is a decimal reduced to a lower denomina
tion .? [64].
6. How are pounds, shillings, and pence changed, ai
once^ into the corresponding decimal of a pound r [66,
67, and 68].
7. How is the decimal of a pound changed, at once^
into shillings, pence, &c. } [70] .
8. What are terminate and circulating decimals }
[71].
9. What are a repctcnd and a perio Meal, a puro
and a mixed circulate ? [72] .
10. Why cannot the number of digits in a neriod bo
equal to the number of units contained in the divisor }
[73].
1 1 . How is a pure circulate or pure repetend changed
into an equivalent vulgar fraction ? [74] .
12. How is a mixed repetend or mixed circulate
reduced to an equivalent vulgar fraction r [76] .
13. What kind of vulgar fraction can produce no
equivalent finite decimal ? [79] .
14. What number of decimal places must necessarily
be found in a finite decimal .? [80] .
15. How many digits must be found in the periods
of a pure circulate ; [82] .
16. When is no finite part found in a repetend, or
circulate ? [S3] .
17. How many digits must be found in tho finite part
of a mixed circulate t [84] .
18. On what principal can we use the properties of
fractions as a means of abbreviating the processus of
multiplication and division ? [86, &c.]
til l<!
176
SECTIOJV V.
PROPORTION.
numbers are given a W]! ^ •T''"' '^ ^^' ^^^^^^^ tluco
found. ^ "' * *^"^*^' ^'»«h ^s unknown, may bo
shown by Hatton, in bis IrthJr i ?^«"''ate, as was
hundred years ago '^^ '''^^" published nearly one
p4oS,^::!l tXs:r^i^T;^ ^^^^ f « ^^
miportant prf clnles nnnn '1/ !? •?, , '^ ^''^^ ^^^^P^e but
i he following tmth3 aro selfevident •_
quantity, 4 for instance Ix^ Id '• vm'^^ ."'"J'"""
equal, we shall have 5X6+4=3X10+^ ' """^ ''™
equal ^':^l^^.:p:''^ij: ^ «"^" 
^. 11 the same, or erjinl r<»..^**
from others whieh a e Tual ^ ''" ''^ '''^''''''^'
equal. Thus, if we subtCt' f." ''"^'^f^^^ ^^^ bo
quantities 7, a'nd S+Cweiu l'.? ""' '' ^'^ ^^^"^^
73=5+23.
And since 8=6+2, and 4=3 + 1.
84=0+2 3+T
PROPORTION.
177
tho golden
' it is termed
. when thrco
wn, may bo
' the simple^
divided into
ate, as was
1 nearly one
tlio rule of
simple but
•0 of ratios.
'i to 0(j[ual
d tlio sum,.
which aro
kvhich are
subtracted
s will bo
the equal
same, or
Thus
if wo multiply the Oi\\\v^h THfl, and 10+1 by 3,
bhall lia\
we
G+(]V3=:U4rix3.
And Kiucu 4 + 0— ID, and 3X().~18.
4+'JX3xG=i;5xl8.
0. If equal quantities aro divided by tlio same, or by
• equtil (uantitit'.s, the quotients will be oqunl. Thus if
wo divide tlie C(iuals 8 and 1+4 by 2, we shall havo
8_4+4
2 T
And since 20=17 + 3, and 10=:=2xr).
20_17+8
10"~~2x5"
7. Ratio is the relation which exists between two
quantities, and is expressed liy two dots ( : ) placed be
tween them— thus o : 7 (reatl, 5 is to 7) ; which means
that 5 has a certain relation to 7. The former quantity
is called tlic onlccedevty and the latter the covscqi/rnl.
S. If we invert the tci ms of a ratio, we sliall havo
their inrenc ratio ; thus 7 : 5 is the inverse of 5 : 7.
9. The relation between two quantities may consist
in one being greater or less than the otlier — then the
ratio is termed arithmctlad ; or in one being some mut
tipk or part of the other — and then it is geometrical.
If two quantities arc ecjual, the ratio between tlunu
is said to bo that of equality ; if they are unequal it is
a ratio of greater inequaUty when the antecedent is
greater than the consequent, and of hsscr inequality
when it is less.
10. As the cviithmetical ratio between two quantities
is measured by their difference, so long as this difference
is not altered, the ratio is unchanged. Thus the ratio
of 7 : 5 is equal to that 15 : 13— for 2 is, in each case,
the difference between the antecedent and consequent.
Hence we may add the same quantity to both tiio
antecedent and consequent of an arithmetical ratio, or
may subtract it from them, without changing the ratio.
Thus 7 : 5, 7 + 3 : 5+3, and 1 >2 : 52, arc equal
arithmetical ratios.
Uut we cannot multiply or dih'ide the terms of an arith
I
if
It
173
PnoPORTlON.
V r
"■■0 c,,uul ; thus 10 : 5=12 : fi L ' • ■ a ^u ","'"
>»t.o by the «u,uo number mthoutaltoring^I.e .aao
U'us 7X2 : VX2=7 : 14bocauso '^^^ 1
tion formerly mvcMi " Wl.^f f... 1 1 f^J" q'les
2U ?" «^l,{i • ' ,. ^*^'^'^ Inichon ot a pound is
7i* , — wJiich in rea ity moans " Wl...f .. / /• •
there between 2\d. and a pound '' or ''WI ? '? ''
consider 2U.. i w. nnn«\r 1' „„^/ ^V hat must wo
or.
If
consider 9iZ It ^^ 1"^""« 5" or " What mu:
Tn fine '?wi;. Ti '""f '^ ''^ P^^"^^^ «« "'"^y ;" »
w ' ^* ^^ ^^>^ value of 2J • 1"
terms by the same numb rffif 1 9 • IZt t^""' "^
ratio a8jf^:«2 or n // ,m, ™ ** ;, ''™ '? 'ho samo
f:4i^,iii,rjj:;°st'^^^^^
•tuuu nmtij rv!»sonta their ratio, aud unity. Thus
PROPOUTION.
179
•n flio liLst example 9 : 000 and ,J^ : 1 arc equal ratios.
H If, not iioc'jssary that wo hIiouKI bo able to oxpiess by
int^igors, nor even by a finite decinuil, what part or mul
tiplo one of the terms is of the other ; for a geometrical
ratio may be considered to exist between any two quan
tities Thus, if the ratio is 10 : 2, 5 ( V) is the quantity
by which wo must multiply one term to make it equal
to the other ; if 1 : 2, it is 05 (^), a fmite decimal : lut
if 3 : 7, It IS M28571' (^), an infiuitc decimal— in which
case wo obtani only an approximation to the value of
the ratio. 13ut though the measure of the ratio is ex
pressed by an mjiiiite decimal, when there is no quantity
which will exacfiy aerve as the multiplier, or divisor of
ouo quantity so as to make it equal to the other—sinoo
wo may obtain as near an npproxunation as we please^
there is no inconvenience in supposing that any one
number is some part or multiple of any other ; tl'at is,
that any number may bo expressed in terms of another—
or may form one term of a geometrical ratio, unity
being the other. "^
14._ Proportion^ or analogy^ consists in the equality
of ratios, and is indicated by putting =, or : :, between
the equal ratios ; thus 5 : 7===Q : 1 1, or 5': 7 : : 9 : 11 (read,
5 IS to 7 as 9 : 11), means that the two ratios 5 : 7 and
9:11 arc eijual ; or that 5 bears the same relation to 7
that 9 does to 1 1 . Sometimes we express the equality
of more than two ratios ; thus 4 : 8 : : G : 12 : : 18 : 36
(vcd, 4 is to 8, as 6 is to 12, as 18 is to 30), m'eana
there is the same relation between 4 and 8, as between
6 and 12 ; and between 18 and 36, as between either 4
and 8, or 6 and 12~it follows that 4 : 8 : : 18 : 36— for
two ratios which are equal to the same, arc equal t'
each other. When the equal ratios are arithmetical, the
constitute an arithiMtiail proportion ; when geometri
cal, a geometrical proportion
15 The quantities which form the proportion are
called proportionah ; and a quantity that, along with
three others, constitutes a proportion, is called a f&iLrth
proportional to those others. In a proportion, the two
outside terms are called the extremes^ and the two middle
terms the means ; thus in 5 : 6 : :7 : S, 5 and 8 are tho
m
J \ ■ A
%•
180
PKOPORTION.
extromos, 6 and 7 the meanf.. mien tlie same qiiantitr
IS found m bolA means, it is called l/ie mean of the
extremes ; thus, since 5 : 6 : : 6 : 7, 6 is tAc mean of 5 and
7. VV hen the proportion is arithmetical, t/ie mean of
two quantities is called their arithmetical mean • when
the proportion is geometrical, it is termed their' ^e^Ts^
onml mean. Thus 7 is the arithmetical mean of 4
and 10; for, since 74=107, 4: 7: :7:10. i\nd8ia
the geometrical mean of 2 and 32 : for, since 5 j
2 : 8: :8 : 32. > ^^^^ s— ai*
16. In an arithmetical proportion, " the sum of the
means is equal to the sum of the extremes." Thus, since
11:9:: 17 : }5 is an arithmetical proportion, 119^:=^
1710 ; but, adding 9 to both the equal quantities, we
have 11 9 + 9=1715 + 9 [3]; and, adding 15 To
n'^'^To h^' •^^^+^+^'^=1^1^+^+15 ; but
H g^.9 + 1^ jge(^^j.j ^^ ii + i5_sinco 9 to be sub
tracted and 9 to bo added =0 ; and 1715 + 9 + i5_
17+9_since 15 to be subtracted and 15 to be added =0 •
therefore 11 + T5 (the sum of the extremes) =17+9
(the sum of the mean ,.— The same thing mi^ht be
proved from any other arithmetical proportion^ and, ■
therefore, it is true in every case. ■ l\
17. This equation (as it is called), or the cqualitv which
exists between the sum of the means and the sum of the
extremes, is the te^t of an arithmetical proportion :— that
IS, It shows us_ whether, or not, four given quantities
corstituto an arithmetical proportion. It also enables us
to hnd a fourth arithmetical proportional to three given
numbers— since any mcc^n is evidently the difference
between the sum of tlie extremes and the other mean •
and any extreme,' the difference between the sum of the
means and the other extreme
A fiT '^r^ •'^r\L? • 11,^^ *^' arithmetical proportioif,9
V ;,+ ^1^^ ' ^°^^' subtracting 4 from the equals,
wo have 1 1 ,one of the extremes) =7+84 (the sum of
the means, mmus the other extreme) ; and, subtracting 7
we have 4+117 (the sum of the extremes minus !no
of the means) =8 (the other mean). V.'e might in the
^amu way nna the remaining extreme, or the remaining
mean. Any othtr arithmetical proportion would hav6
PROPORTION.
18i
answered just as well — hence what we have said is true
in all cases.
18. Example. — Find a foiirfa proportional to 7, 8, 5.
Making the required number one of the extremea, and
Cutting the note of interrogation in the plac"e of it, we have
: 8 : : 5 : '? ; then 7 : 8 : : 5 : 8}57 (the sum of the means
minus the given extreme, =6) ] and the proportion com
pleted will be
7 : 8 :: 5 : 6.
Making the required number one of the means, we shall
have 7 : 8 : : '? : 5, then 7:8:: 7+58 (the sum of the
extremes minus the given mean, =4) : 5 ; and the proportion
completed will be
7 : 8 : : 4 : 5.
As the sum ov the means will be found equal to the sum
of tlio extremes, we have, in each case, completed the pro
portion.
19. The arithmetlad mean of two quantities is half
t\\Q sum of tho extremes. ¥oy the sum of the means is
equal to the sura of the extremes ; or — since tho means
are equal — twice one of the moans is equal to the sum
of tho extremes ; consequently, half tho sum of the
means — or one of them, will be equal to half the sum of
the extremes. Thus the arithmetical mean of 19 and
(=23) ; and the proportion completed is
27 is
2
19 : 23 :: 23 : 27, for 19 + 2^=23423.
20. If v/ith any four quantities the sum of the means
is equal to the sum of the extremes, these quantities aro
in arithmetical proportion. Let tlio quantities bo
8 7 5.
As the sum of the means Ls equal to tho sum of tho
extremes
8 + 5 = 0+7.
Subtracting 6 from each of the equal quantities, wo
have B+fv— 6 = 6 + 7— 6 ; and subtracting 5 from each
of these, we have 8 + 565=6 + 76— 5. But
8 + 5=6 — 5 is equal to .R — 6, since 5 to be added
and 5 to bo subtracted are ?=;0 ; and +6 + 7—6—0 =
7—5, since 6 to be added an^l 6 to '/)Q subtracted =0 ;
I 2
.82
PROPORTION,
V' m
tlierefore 8+5 — 6 5 R47 p, r. • .l
fortion. It might in the same way be noyed Ui»i
«y e^te ibnr quantities are in arithnfeacal^p^I^orUon
21. In A gamttrical proportion, "the jiroduct nf
t^'J fulfil ''' ^6* 8 IS a geometrical proportion,
L~hv 7 2\ ' ^u tiplymg each of the equal quantl
nes by 7, we have (V»X7^ — 'Jv7. onri ^ u i •
.aehofthe.eby8,weUelfef6x'7(ox7''P'r^
•ut 14X8 13 the product of the extremes and 1 6^7
B he product of the means. The same"eLonTn« lull
r" rs^\ri;ter«' '''»^'> ^^
Mt^a^r^rpfet^^^^^^^^
7X^S2lVxnV iV f •^. *''^g^°»«*™'J proportion,
/ X ^—HX n i and, dividing the equals by 7, we hayo
32 (one of the extremes) =1*^ (the product of the
mo^ns divided by^the other extreme) ; and, dividing these
by ll,wehayejj(the product of the extremes di
vided by one mean)=14 (the other mean). We miBht
l^l^Zli t 'l^' proportion would have answered
just as well — and thfirfifmo wKo* l__. _ i . :
in every case. ^ """" "'' "*'" ^^"^ ^ ^^^^
PROPORTION.
ibH
mo as
5, are
itSS. Example. — Find a fourth proportional to 8, 10, and 14.
Making the required quantity one of the extremes, we shall
10X14
Lave 8 : 10 : : 14 : ? ; and 8 : 10 : : 14
8
(the product
of the means divided by the given extreme, ==175).
And the proportion completed will be
8 : 10 : : 14 : 175.
Making the required number one of the means, we shall
8x14
have 8 : 10 : : ? : 14 J and 8 : 10
10
(the product of
the extremes divided by the given mean, =112) : 14.
And the proportion completed vnll be
8 : 10 : : 112 : 14.
£X£RCISf:8.
Find fourth proportionals
1. To 8,
6.
6.
6, and 12
8
6, 12
10, 150
1020, 68
160, 10
68, 1020
68
150
1020
10
Jlns. 24.
16.
1020.
10.
68.
160.
24. If with any four quantities the product of the
means is equal to the product of the extremes, these
quantities are in geometrical proportion. Let the
quantities be
5 20 6 24,
As the product of the means is equal to the prod. !t
of the extremes,
5x24=20x6.
5X24 20X6
Dividing the equals by 24, we have""^^ — = 24 ' J
5X24 20X6
and, dividing these by 20, we have 20X24
But::^
5X24
_5 20X6
=20 5 and 20x24'
20X24
_6^ 5 _6 ^
=24' j therefore 20 ~ 24 *
20X24
consequently the geometrical relation between 5 and 20
two equal geometrical ratios — or a geometrical propor
184
PROPORTION.
the pTodJotTelnZll «°™r'"f proportion, I
eztromes. °^ '^ '^l""' '» ">« P'oduot of the
proportljfnal^""' '^'' '^™ '^ ™"y. '> «■><• a fourth
a{i;«._Knd the product of the second and third
Example, — Whnf i'<a +1,^ ^ ^ iuira.
2^ ^ vvhat IS the fourth proportional to 1, 12, and
W ^ • ^^ • •' 27 : 12x27=324
«nee dividing a nui fy "^u/dt^Tot'il?" u'^ '"^^
EXKRCISES.
Find fourth proportionals
,XTol, 17, and 8
}?• » J' 23 „ 20
J » J. 53 „ 110
^*» •• I, 15 .. 1234
^n*,
j»
136.
460.
7300.
6830.
18510.
by the first. ' ""^ "^ *''^°' ''iM' is not unity
Ex.«P„.Find a fourth prop„ra„„„ to 8, 1, „ud 5.
4"^ut'it':ftL^;:etilV, "."yt"" given
product of both, ™hen tl'e otC hZu^ "JT^dered ^ tho
pit,on .y „„H, p,„auce. no Sn^V?; Sr'""
EXERCISES.
Find fourth proportionals.
fi on „.. 1
H. To
15. „
16. ,.
17.
18.
19.
20.
21.
5.
6,
7,
8,
6,
37,
^?00, 1000
200, 1
20, and
1
21
24
1
1
1
20
1
1
50
68
1
1000
^?i
4
4.
3.
8.
Si.
4.
6.
6.
= ' ^"^^or tiio extremes; and the proan^"
ROLE OF PROPORTION.
185
of the extremes is equal to the mean multiplied by itself.
Hence, to discover the geoinclriccl mexin of two quan
tities, we have only to find some number which, multi
plied by itself, will be equal to their product — that is,
to find, what we shall terra hereafter, the square root
of their product. Jhus 6 is the geometrical mean of 3
and 12; for 6X6=3X12. And 3 : 6 : : 6 : 12.
28. It will be useful to make the pupil acquainted with
the following properties of a geometrical proportion —
We may consider the same quantity either as a mean,
or an extreme. Thus, if 5 : 10 : : 15 : 30 be a geometrical
proportion, so also will 10 : 5 : : 30 : 15 ; for we obtain the
same equal products in both cases — in the former, 5X
30=10 X 15 ; and in the latter, 10 X 15=5X30— which
are the same thing. This change in the proportion ia
called inversion.
29. The product of the means will continue equal to
the product of the extremes— or, in other words, the
proportion will remain unchanged —
If we alternate the terms ; that is, if we say, " the
first is to the third, as the second is to the fourth" —
If we " mnltiplijy or dimk the first and second, oi
the first and third terms, by the same quantity"—
If we " read the proportion badcwards''^ —
If we say " the first term plus the second is to the
second, as the third plus the fourth is to the fourth"—
If we say " the first term plus the second is to the
fii'st, as the third plus the fourth is to the thiid"— &c.
RULE OF SIMPLE PROPORTT.ON.
3D. This rule, as we have sr.id, enables us, when threa
quantities are given, to find a fourth proportional.
The only difficulty consists in stating the question ;
when this is done, the required term is easily found. _
In tlie rule of simple proportion, two ratios arc given,
the one perfect, and the ether imperfect.
31. IluLE— I. Put '^^hut given quantity which belongs
to the imperfect ratio in .u third place.
II. If it appears from the nature of iho question that
the required quantity must bo greater than the other,
J86
RULE OF PROPORTION.
■!!m !
4%o~};^aY^^Tir^^^^^^ 7^3 fa wall in one
It will faoiKrt Jb aLi Tth« n 'm *^^'T' ""^'^ '
question briefly, as fo lowsuini JZF^}^ ^^^^ ^^^'^ *''«
represent the required qlLti^yl^ "^ ''"*' '^ interrogation to
5 men.
10 yards.
21 men.
'Jyai'da.
^r^^ps!z a;'iat ^■"'^^^^^^^ '^*^^^ ^.
P/Jratio7an^d\^S3^,r"^*^*^^^ ^^»«h form the
than 5 men the VpnmW "'i*^ ^ §'^'^*«'' ""'"^^r of yards
than the^";, nu Jber~h«^^^^^^^ ^^'^' ^^" ^« S'^ater
term of the peSt ra^n ^n Ji,'° **"'' ^^ ^' ^° P"<= the larger
the first plac?!I '^ '"^ *^° «^'°°'^' «^d *he smallerlu
A ^ 5 ; 21 ; : 10 : ?
And, completing the proportion,
5 : 21 : ; 10 • ^1 X 10 .o ^u .
■ ~~5 =^A *he required number.
last 5 .en . m: tSt'^:^'Z,^g ?„/■ 8". i. to
3 men.
2 days.
5 men.
*? days.
tiitrbV^u^fiTher^ii'"^"^"' ""«'^« ■»''
„„ "'t^.'rs?,'' 'h" nnmbor of men, the shorter «.„ ,;„» „ .:..._
4««nt..j, „, .,re,ui Will last thorn; but th«i',»7,^ «— g'Ji^
RULE OF PROPORTION.
187
required quantity — henco, in t'his case, the greater term of
the perfect ratio is to bo put iu the first, tmd the smaller in
Uio second place—
5 :3::2:'?
And, completing the proportion,
=1, tlie required term.
5:3::2
5
34. Example 3. — If 25 tons of coal cost £21, what will
be the price of 1 ton '?
25 : 1 : : 21
pounds £.jp=lG5. 9^f/.
25
25"
It is necessary in this case to reduce the pounds to lower
denominations, in order to divide them by 25 ; this causea
the answer, also, to be of different denominations.
35. Rkason of I. — It is convenient to make the required
quantity the fourth term of the proportion — tliat is, one of the
extremes. It could, however, be found eqxially well, if conHi
dered as a mean [23].
Urahon of II. — It is also convenient to make quantities of
the same kind the terms of the game ratio ; because, for in
Btancc, wo can compare men with men, and days with days — >
but wo cannot compare 7ne7>. with days. Still thero is nothing
inaccurate in comparing the number of one, witli the number of
the other ; nor in comparing the number of men with the quan
tity of work they perform, or with the nximbtr of loaves they
eat ; for these things are proportioned to each otlier. Hence wo
shall obtain the same result whether we state example 2, thus
6 : 3 :: 2 : ?
or thus 5 : 2 : : 3 : ?
When diminishing the kind of quantity which is in the per
fect ratio increases that kind whicli is in the imperfect — or the
reverse — the question is sometimes said to belong to tlie inverse
rule of three ; and different methods are given for the solution
of the two species of questions. But liatton, in his Aritli
mctic, (third edition, London, 1753,) suggests the above gene
ral mode of solution. It is not accurate to say " the inverse
rule of three" or " inverse rule of proportion ;" since, although
there is an inverse ratio, there is no inverse proportion'.
Reasoiv of III. — We multiply the second and third terms,
and divide their product by the first, for reasons already given
[22].
The answer is of the same kind as the third term, since
iicitiici' tiic iiiullipliuukiuu, iiur aixxi UiTinluii Oi inia ici lii ixixa
changed its nature ; — 20*. the payment of 5 days divided by 6
188
RULE OF PROPORTION.
Of J da, ,„„u,p,iea b, 9 givo» ^ x as the pa,„o„t of u
would not to the 4u^;a'';„;X;i^« «;»;,<' tl,are.ore ij
scc'oL,^«"fiLfa„°d'','i,i?,T"™\'" 'i'""^" *'"> «'■»' ""•'
mo„ m'ca ure?wbe\ tW o""'' ''^ ""='■• ^■"■•"'=«' <«"■'
<itlier [29J. ^ '"'"" "'o ""'"poato to each
Ex.„P..._,f 30 cwt cost ^24, what «„ 27 owt. co»t '
Dividing the first and second by d we have
And, dividing the first and'third" by 4,
EXERCISES FOR THE PUPIL.
^ ^^"J a fourth proportionalto
4. 6 yard, : I yard : : 27,, Am. 4,. Cul
b. 5 lb 1 ib : : 155. ^,„ g^
7. 4 yard, : iSyards : : u. Am. 4s. 6d.
J? ^t Ssf *»^^^^^^ii^^^o to at £25 p„r
p!ooe/eo:"?'Z."'^?!;;S "°^' ^2^. '«w much will 50
Ans. 121 „,"„(?„ ' '""° "■""" "'«y »"ffi» for 32 r
■ cwt"; 'aT. JIH^' ''■ °f "^J" .St at 50. PC,
«.o latter shall I ro.;iror''l:"2o";ardV'"" "'""'' "'
the paymont
payment of U
ns the third,
therefore it
lio first and
iatosfc coni
ite to each
IS. £270
£23 per
I will 50
i last 40
for 32 :
50.y.
poj
h wide,
iiucli of
RULE OF PROPOUTION.
189
, ,13. At 10.?. per barrel, what will be tho price of 130
barrels of barley ? Ans. £Q5.
14. At 5s. per lb, what will be the price of 150 ft) of
tea ? Ans. 7505.
15. A merchant agreed with a carrier to bring 12
cwt. of goods 70 miles for 13 crowns, but his waggon
being heavily laden, he was obliged to unload 2 cwt. ;
how far should he carry the remainder for the same
money ? Ans. 84 miles.
lo. What will 150 cwt. of butter cost at £3 per cwt, }
Ans. £450.
17. If I lend a person ^£400 for 7 months, how much
ought he to lend me for 12 > Ans. £233 6s. 8d.
18. How much will a person walk in 70 days at tho
rate of 30 miles per day .? Aiis. 2100.
19. If I spend £4 in one week, how much will I
spend in 52 ? Ans. iS20S.
20. There are provisions in a town sufficient to sup
port 4000 soldiers for 3 months, how many must bo
sent away to make them last 8 months ? Ans. 2500.
21. What is the rent of 167 acres at £2 per acre ?
Ans. £334.
22. If a person travellmg 13 hours per day would
finish a journey in 8 days, in what time will he accomplish
it at the rate of 15 hours per day > Ans. 6f days.
23. What is the cost of 256 gallons of brandy at 12s.
per gallon ? Ans. 3072s.
24. What will 156 yards of cloth come to, at £2 per
yard .? Ans. £312.
25. If one pound of sugar cost 8^Z., what will 112
pounds come to .? Ans. 896d.
2b. If 136 masons can build a fort in 28 days, how
many men would be required to finish it in 8 days }
Ans. 476.
27. If one yard of calico cost 6^., what will 56 yards
come to ? Ans. 33bd.
28. What will be the price of 256 yards of tape ak
2d. per yard ? Ans. 612d.
29. If £100 produces me £6 interest in 365 days,
what would bring the same amount in 30 da^'S .'' Ans
j&i:<5iD iJi. 4a.
'♦i
,f
i
m
14*0
nuLE OK I'noromio.v.
30. What shall I receive for 157 pair of gloves, at
I Of/, per pair? Ans. 157 Od. fa > «*"
31 What would 29 pair of shoes como to, at 9* ner
pair? Ans.2Gls. v., »t ^j. per
aJ^: / "^ ^TT ^'°^, ^"' neighbour a cart horse which
draws lo cwt. for 30 days, how long should he have a •
horse m return which draws 20 cwt'? Ans. 22i lys
rr\l%a "''"' P'\*^ ["^ '"*«'''''^ ^<^ ^'^ V^r cent." would
give £6 m one month ? Ans. ^£1200
34 lfllendi2400for 12 months, how lone; our^htJEl. 50
be lent to me, to return the kindness ? Am. 32 months
hsflO^orS' 'V ^^^^"^^"/'^re found sufficient to
last 10,000 soldiers for 6 months, but it is resolved to
add as many men as would cause them to be consu.hcd
Anl 20 000. ' ''""'^'' '^ ™'" '""''^ '^'^ ^^"^ ^^ ^
/•n.^o ^^ ^,,^°7<^''' subsist on a certain quantity of hay
for 2 months, how long will it last 12 horses ? A^^
1} months.
n/i^'/ '^^^P^^f'eper is so dislionest as to use a woi<rht
of U for one of 16 pz. ; bow many pounds of just v
be equal to 120 of unjust weight ? Ans. 105 lb
rlnw • r' ""^ ^^^ **" ^^ '"^^^^^ ^J 40 men in 10
Ans.Vs^ days.""''^^ ""'^^^ '' ^' ^'''"^''^ ^^ ^^ "^^° •'
are^Lt^IfTl *''' ^''^^'^ '''^"^^ ^'™' ^^^^^^ proportion
are not of the same denomination ; or one, or both of
them contain different denominations—
HuLE.— Eeduce both to the lowest denomination con
tamed m either, and then divide the product of the
second and third by the first term.
pomrcoTt /"^^ '^''" '''''''' '^ '''' '''' 1^^^ ^vh'^t ^ill 87
The lowest denomination contained in either is ounces.
':■. ?_ '■ 1302X15 d.
 ^ • j^J • • 15 : 3— =6960=£29.
1392 ounces.
There is evidently the same ratio between 3 oz and 87 Th
as between 3 oz. and 1392 oz. (the equal of 87 ft)
RULE OF rnOPORTION.
101
ExAMPi.E 2.— If 3 yards of any thing cost 4^. 0J(/., what
can 1)0 bought for £z i
The lowest denomination in either is farthings.
s. d.
4 9?
12
57 ponce.
4
20 231
40 shillings.
nls.
3.
231 farthings. 480 pence.
1920 farthings.
There is evidently the same ratio between 4*. ^Id. and /2,
fls between the numbers of farthings they contain, respectively
For there is tlio same ratio between any two quantities, us
between two others which are equal to them.
Fa'amplk 3.— If 4 cwt., 3 qrs., 17 lb, cost XIO, how much
will 7 cwt. 2 qrs. cost ?
The lowest denomination in either is pounds.
f' 840x10
19 : • ^^,^ =£29 Is. bd.
cwt.
qr.
lb
cwt
qr.
4
3
17
: 7
2
4
4
19 (
^rs.
30 <
:r8.
28
28
549 lbs.
840 K)3.
EXERCISES.
Find fourth proportionals to
39. 1 cwt. : 17 tons : : £5. Ans. £1700.
40. bs. : £20 : : 1 yard. Ans. 80 yards.
41. 80 yards : 1 qr. : : 4005. Am. Is. 3d.
42. 3s. 4d. : £1 10s. : : 1 yard. Atis. 9 yards.
43. 3 cwt. 2 qrs. : 8 cwt. 1 qr. : : £2. Ans. £4.
44. 10 acres, 3 roods, 20 perches : 21 acres 3 roods :
£60. Ans. £120.
45. 10 tons, 5 cwt., 3 qrs., 14 ft : 20 tons, 11 cwt ,
3 qrs. : : £840. Ans. £1680.
109
i.ja*
RtTLE OK PROPORTION.
cwf ? ^t' 'lltoo'"^° '' '' "^^ ^' '^"^^' '' "^^ P
pricL?i5irr'if ^*« ''> Hat will bo tho
yar^dt inri ^nn^^/ •Z"*^ '°'*' ^^ ^' ^^^* ^iU no
17 .tf o °^* ^^^^""er costs ^26 6^., how much 4ill
^7^ 8^1 cwt ^'' '"^"^ ''^''' can I have for £615 isi."?
57. How much beef can be bought for £760 12* al
1 ,1 i ^^' ^ ^''•' "^ ^^*' cost £150, what will 3 ft,
1 oa., 1 1 dwt., cost ? Ans. £37 105. '
69 If 10 yards cost 17.., what will 3 yards 2 ars
cost? Am. bs. Uid. » j'»iUH, * qrs.
60. If 12 cwt. 22 ib cost £19, what will 2 cwt ^
qrs. cost ? Am. £4 5*. 8^^. "^ ^m ^ cwt. 3
n^nV 14 ^^ ""'•'/? ^7*' ^^ g"'> «««* 19*> what will
13 oz. 14 grs. cost ? Am. 15s. lOd.
mination ' ^^''^ *°''"' '°'''''*' °^ "'^''^ *^'"^' ^'^« ^cno
if "^rf •■""■^fi!^"''^ '^*? ,*^^ ^^^^'^* denomination which
contains then multiply it by the second, and div de
the produc by the first term.The answe^ wfll be of
hat denommation to which the third has been reduced
u^ rnay sometimes be changed to a higher [Se^
RULE OF I'ROPORTION.
19S
Example 1.— If 3 yards cost ds. 21(1, what will 327 yards
Tho lowest denomination in the third terri is farthings.
yl" ^i" *• i\ 3'^7x441 £ s. d.
3 : 327 ; : 9 2 : ^ farthing8=50 1 6.
12
^)i. 110 pence.
4
441 farthings.
Kx AMPLE 2.— If 2 yards 3 qrs. cost 11 W., what will 27
yards, 2 qrs., 2 nails, cost 1
Tho lowest denomination in the first and second is nails,
and in the third farthings.
yds. qr. yds. qr. n.
2 3 : 27 2 2
4 4
d.
lU
442x45
— 4^ — farthing8=9<. bd.
11 qr. 110 qr.
4 4
44 nails. 442 nails.
45 farthings.
Reducing the third term generally enables us to perform the
required raultiplicatiou and division witli more facility. —It ia
sometimes, however, unnecessary.
Example.— If 3 lb cost £3 lis. 4\d., what will 96 lb cost?
n> lb £. s. d. ^ s. d. £ s. d. £ s d
3: 06:: 3 11 4; : ' '^"^ =3 11 4Jx32=114 4 8
EXERCISES.
Find fourth proportionals to
62. 2 tons : 14 tons : : ^228 10*. Ans. 199 10*.
63. 1 cwt. : 120 cwt. : : 18^. 64. Am. .£111.
64. 5 barrels : 100 barrels : : 6s. Id. Ans. £6 Us. Sd
. 65. 112 ft) : 1 ft) ; : ies 10s. Ans. l{d.
r66. 4 ft) : 112 ft) : : b\d. Ans. \2s. 3d.
67. 7 cwt., .3 qrs., 11 lb : 172 cwt., 2 qrs., 18 ft) : : £,3
9s. A\d. Ans. £87 55. Ad.
n ;""!i
194
RULE OF PROPORTION.
68 172 cwt., 2 qrs., 18 lb : 7 cwt., 3 qrs., 11 lb : : ^87
6*. 3^^. A71S. £3 195. 4id.
Am'Jl ^^^•' ^ ^^■^•' I'* * * 2 cwt., 3 qrs., 21 lb : : £73
70. £87 Gs. 3d. : £3 19s. 4i^. . : 172 cwt., 2 qrs., 18
ib. Ans. 7 cwt., 3 qrs., 11 lb. > ^ '
71 £3 195. 4irZ. r £87 65. 3r/. : : 7 cwt., 3 qrs., 11 lb.
Am. 172 cwt., 2 qrs., 18 ib. > ^ »
/l^^^'^tll^^^" ^'^' ^^"^ ^^*>^^a* ^ill 120 cwt. cost.?
^.L' £1^05*^4/" ''""'' ^^^''^"^ ''^ ^ ''^' '''
74. What will 120 acres of land come to, at 145 6d
per acre.? ^w5. £87. ' '
75._ How much would 324 pieces come to, at 2s S^d
per piece ? Ans. £43 175. 6^/. ' f '
76. Whafr is the price of 332 yards of cloth, at I65.
4^/. per yard .? ^7*5 £107 1 65.
i/^" ■'•^ l?^^^^ ^^ ^^''"^^ ^^^^^ ^'' 4d!) what will 18 lb
10 oz cost.? Ans. £49 135. 4d
t. ?'L'. £1^2 13?4f ' "'^^ "'^ ' ^"^ ^ ^^ 
rent ( I 156 acres 3 roods .? Ans. £089 I45.
^fl" ^^* }^^' ^'^' P^^ ^''•' what will 56 cwt. 2 qrs bo
worth.? Ans. £118 13.S. ^
81. At 155. 6^ per yard, wliat wHl 76 yards 3 qrs
come CO .? Ans. £59 95. 7id ^
lb ?' 2' £r065.'' """'' ' "^ ''"' *'' ^* ^'' ''• P«^
83 At 145. 4d. per cwt., what will be the cost of 12
cwt. J qrs. .? Ans. £8 195. 2d.
84. How much will 17 cwt. 2 qrs. come to, at 195.
lOJ. ^er cwt. A71S. £17 75. Id.
2 n?; "^i"""* °^^^"«^«osts £6 65., what will 17 cwt ,
2 qrs , 7 lb, come to .? Ans. £102 125 lOi^^ '
■;■ .^IJ ^^ ^'^ ^ cost ^^; lo.v. 9^ , Avhat will be the
cost of DO cwt., 3 qrs, 24 Jb .? Ans. £378 I65. 8^1
RULE OF PROPORTION.
195
87. If tlic shilling loaf weigh 3 ft 6 oz., when flour
sells at £1 13s. 6d. per cwt., what should be its weight
when flour sells at £1 7s. 6d ? Ans. 4 lb 14f oz.
i,£8. If 100 lb of anything cost .£25 Bs. 3d.,\lmt will
be the price of 625 lb ? Ans. £WS 4s. 0^~d.
S9. If 1 lb of spice cost 105. Sc^., what is half an oz.
worth > A'ns. Ad.
90. Bought 3 hhds. of brandy containing, respectively,
Gl gals., 62 gals., and 62 gals. 2 qts., at Qs. Sd. per
gallon ; what is their cost.? Ans. £Q1 16s. 8d.
39. If fractious, or mixed numbers are found in ono
or more of the terms— •
KuLE. — Having reduced them to improper fractions,
if they are complex fractions, compound fractions, or
mixed numbers— multiply the second and third terms
together, and divide the product hy the first — according
to the rules already given [Sec. IV. 36, &c., and 46.
&C.J for the management of fractions.
Example.— If 12 men build 3^ yards of wall in ? of a
week, how long will they require to build 47 yards 1
Sf yarJs=2,6 yards, therefore
. . i^X47_,
26
1
47
7
=9] weeks, nearly.
"0. — If all the terms are fractions —
lluLE.— Invert the first, and then multiply all the
terms together.
ExAMPLK.~If f of a regiment consume \l of 40 tons of
flour in  of a year, how long will ^ of the same regiment
tako to consume it ?
i'lV f XlT=^XfX?=,^=2028 days.
Tin's rule follows from that which was given for the division
of one fractiou by another [See. IV. 49].
41. If the first and second, or the first and third
terms, are fractionsr—
Kui.E. — llediuie them to a common denominator
(should they not have ono already), and then omit tho
denominatorsi
i '
S^K i
li(f
\m
'Ji:li
RULE or PROPORTION.
a o^^t'cS"" ' "^ ^ ™' "^ ""' «™'' •^2, what ,iU ^ of
I : .J . . 2 : "J
Reducing the fractions to a common "denominator, we have
, , . fff •• U :: 2:?
And omitting the denominator,
20:27::2:2^=£27=£2 14,.
andVhiiH?®'"^^^^.""".^.*'^^^^"^ *^® fi'^t and second, or the first
BO^it^tTopitr"'^^" denominatorwhiclfc^S^I Zll
EXERCISES.
cos'ti' ^Aii'ijr '"*' '"' "^^' ^"^ ' ^ ^^ •
aI%^''' """'^ ""^^ ^ ^"'^ '^^^ *« if 1 «ost is. ?
of 'S^'^t^^'T^ ^^^^ ^^^^ ^'^' ^« *^^ !>"
^r'^7?f 4T]' '« ^^ «^ «^^^ «^ ^' 6i per oz. ..
I h'avo V^l^f^'^Tk^f^ '^" "'^°^ P™^^ «
cosf ^7^«f i!^ *^ ?"'^ ^^ ^^T=V yards of cloth, if 7f
cost i^7 lb5. 4d. .? ^W5. iE51 35. 113 3^ ' •
R. ^??;7 ^l ^^^^.vi '^^^^ '' ^^'*^ ^981, what will ie363
85. 7^d. be worth .? ^?w. ^^^358 7, ij' ''^'*
bought for ^i2'3''pP'l^ for 4 yards, how much can bo
Dougttt tor £2j\ > Ans. 24 yards, nearly.
MISCELLANEOUS EXERCISES IN SIMPLE PROPORTION.
102 Sold 4 hhds. of tobacco at 10ifZ ner TK • INTn 1
weighed 5 cwt., 2 qrs. ; No, 2 5 ^™' V •_ P , ^« « ^
iTirf
1 ^.. 1 /< K
•»T
lb ; and No. 4, 5 cwt., 1
^j'., X i lu : ISO.
pnoe.? ^7M £]04 Us. 9d
, 1 qr., 21 lb. What
RULE or PROPORTION.
197
1 03 . Suppose that a bale of merchandise weighs 300 Jb,
and costs £15 45. 9d. ; that the duty is 2d. per pound ;
that the freight is 255. ; and that the porterage home
is Is. 6d. : how much does 1 ib stand me in ?
£ s. d.
15 4 9 cost.
2 10
1 5
1
ft)
300
lb
1
duty.
freight,
6 porterage.
: 19
20
•iOT
12
1 3 entire cost.
300)4575
15d. Answer.
104. Heceived 4 pipes of oil containing 480 gallons
which cost 55. 5^d. per gallon ; paid for freight 45. pet
pipe ; for duty, 6d. per gallon ; for porterage, l5. per
pipe. What did the whole cost ; and what does it stand
me in per gallon > Ans. It cost £144, or 65. per gallon
105. Bought three sorts of brandy, and an equal
quantity of each sort : one sort at 55. ; another at 65. ;
and the third at 75. What is the cost of the whole —
one gallon with another ^ Ans. 6s.
106. Bought three kinds of vinegar, and an equal
quantity of each kind : one at ^^d. ; another at 4d. ;
and another at 4Ji. per quart. Having "mixed them
I wish to know what the mixture cost me per quart }
Ans. Ad.
107. Bought 4 kinds of salt, 100 barrels of each ;
and the prices were 145., I65., 175., and 195. per barrel.
If I mix them together, what wOl the mixture have cost
me per barrel } Ans. I6s. 6d.
108. How many reams of paper at 95. 9<i., and
125. 3d. per ream shall I have, if I buy £55 worth of
both, but an equal quantity of each .? An,s. 50 reams
/\T an on
109. A vintner paid £171 for three kinds of wine :
one kind wa,s £8 IO5. ; another £9 55. ; and the third
'if;
lUS
%!'
RULK OF riioroRTroN.
Iia.l of
•fiJO l')s, nor hjii? TL. I,., j r» i
♦'''•'' orchid. ^^^^^^^^^^^^
10 15
28
10
2H 10, the prioo (.f throo J.og.shoadH of oa.,h
£
171
, X171x;{
■ £2H 10~^^ ^'''Js.
I)arr,I.s had I „f oach > yi,,, yoo ■*'•"• ^'"'■f '"•■"ly
weeks. "^^ weoiv ^ Ans. 56
provisions. Ifow lot' w^l/'^'^?^^^*'' ^^''^««« ^^^ ^>f
and 2 ^^y^, ^ ^''"S ^'" *''^^3' ^^^ ^ ^ns. 26 weeks
page. At .'],at ,,e n^I^ Te eVpo'ct'uo ho" '?''^^
copy contaming 400 pa/s P ^;.'^; %f;«"j;n «
il^^^.y^^Z^7 'V^^^ "«'«b^r of 'each:
117. Suppose that a i^reyhoiiiul molroc 07
>vlnle a Jmre makes O;! ?nd h 1. • '^ 'P''"'S^
rqunl Icii.r(h T„ i„„; ,' ^"'^'^ '''^"' «P'"ings are of
«^^'t.ikc.i, u .he IS au .prmgs before (he hound ? '
RULE or riiorouTiON.
The tinio tukc^n by tlio gic^huuiid for ono
that ioquinMl by ihc hnw., ah 2') : 27
as 1 : n
iJ9
Bprin/i; in to
^^^ UJ 'I'J'u ^';r('yliomi(l, ilicn'iuio, iraiiiif "'.^p of
OV U8
Hpnng Uunug ovcry ^prin;,' of (1h! luirc. 'Jlicrol
oro
tl»c
: 50 : : 1
liavo will m
snriii;^; : 5()^^",=rzG7^), Ihc number of Hprinjra
ako, bi'l'oro it is overtake:!.
10
118.^ If a tun of tallow oo.sts ,£35, nnd iH sold at tl .
rate of 10 per cent, profit, what in tho solliinr prico >
Ans. JU3S U)s. °
119. If a ton of t;ilIo\v costs ,£.17 10.y., at what rntt
muMt it bo Hold to gala by U) tons tlio price of I ton >
Afis. £40.
120. JJought 45 barrels of boof at 21. v. per barrel;
auion,!^ tliem aro IG barrels, 4 of whi(3h would bo wortii
only li of tho rost. Mow muoli must I pay t Ans.
£43 l.v. ^ ^
121. If 840 oggs aro bought at the rate of TO for a
penny, and 21G more at 8 for a peiniy, do I lose or gain
if I soil all at ly for 2d. ? Avu. I gain ikl.
122. Suppose that 4 men do as much work as 5
women, and that 27 men reap a (juantity of corn in Mi
days. In how many days would 21 wonum do it .? Ans.
Tlio work of 4 mcn=that of 5 women. Thorefore (divldinj;
each of tho equal quantities by 4, they will remain e(pialj^
4 men's work . , s <hu work t)l'5 women
^ (one mans work )=  ■ ^ . Con
bcquently Ij times tho work of one woman=rl man's work .^
that is, tho work of oiio man, in t(U'ma of a woman's woik,
is 1{ ; or a woman's work is to a man's work :: I : 1'.
Hence 27 mens work = 27xl womon"s work 3 then, in
place of Haying —
21 women : 27 men : : 13 days : ?
say tho work of 21 women : the work of 27xU r=33n
3;J''xl3 ^ '^
%=:20^« days.
women : : 13.:
_ 123. The ratio of the diameter of a circle to its
circumference being that of 1 : .JMlf)!), what is the
circumference of a circle who.si! di;inu^ter is 473G feet ^
Ans. 14878018 feet.
124. If a pound (Troy wrllit) of .silver i,s worih (JGs.,
!H\
200
RULE OF PROPORTION.
whaj^is the value of a pound avoirdupoise ^ Ans. ^
ere^ltttaX"^^ ^'^^^^ to his
16.. 35^. '''''^ ^'^ ^^^* <^a" ie pay .? Ans. £m
iei347 // t^^'feSl Z i"^"^"^^i^ «^ 1 costs
Am. £1714 ihlUld ' *^' '°'* ^f ^« Irish mile.?
127. If the rent of 46 aprpq q ,.««j i
>s JeiOO, what will be the re^t'of '^^' ^"^ " P^oh^s,
10 porches? Ans. ^vZTef, '' '"''"^' ^ ''^' ^
12 mL aday^B wl!: Sf'" .f/"'^^ "' '""e rate of
him. How iy mUea a dir^'i^'p*^ '^y'' "^''^"k
both to have started C„ ,hfl ? "■*^<'' *"»™g
129. If the TOlue of Tn^ T° ^■^'f" ' ■^'" 17.
^£4 0,. 2j/ how manvTir avoudupoise weight bo
Pomd1roy'>AmclS' '^y ^o tad & one
anf S;e\wf:batt:»t\" ' ^^'"'"« '" "« n'!
'o' whaV : Se'w:!':?!:' \^'''™ " »» ^
jeiO 7s. 8J^. "'^ ""« ^^''ols garden? ^^j.
;> ef da^s raXrii'dt" i:t ? '* ■^^^f ' «
three do it ? A71S. 2~i2 ^ ^^^* *^^® ^o«ld all
« ^a,s , 1 aa, , , ,,,■ ^^^ ^^^^ _ ^ ^^^^ ^^^^^ ^^^^^
«i<^. = lCa, = : lwi,o,o„ft.e;7k'r^fp!rntro&:;
#43 1. 3 __i44 7 °^7^^<^ C would do in a day
finished in ^ davk'Tn"!*!'';*. .">"»""=■■' ''^l •>«
"We to do U by hbuself?" ^i^i. Jr^Y^'^ ""»"f « 1«
' ^ Atis. de4
1871 to his
'^212577517
? ^%*. ^£30
' canal costs
Irish mUe ?
14 perches,
J roods, and
the rate of
's, overtook
5l:i allowing
47W. 17.
weight be
id for one
lis tenant ;
Dies to £4
1 ^ Am.
days; B
would all
le whole —
3 in a day.
le whole —
> in a day.
e whole —
> m a day.
in a day.
ivrork :: 1
le work) :
Fit.
B in 6^
'• will be
d C be
RULE or PROPORTION.
201
• ■ A, B, and C's work in one day=£ of the whole=jJ
Subtract j A's work in 1 day=JV I _i i o of tha whole **»
ing j B's work in 1 day=/j. j ^s^ ^^ *^® wnoiey^^,
C'8 work in one day remains equal to . . . ^^^^
Then f^}^ (C's work in one day) : 1 whole of the work : : 1
day : 2 ^i, the time required.
134. A ton of (Jbals yield about 9000 cubic feet of
c;as ; a street lamp consumes about 5, and an argand
Murner (one in which the air passes through the centre
of the flame) 4 cubic feet in an hour. How many tona
of coal would be required to keep 17493 street lamps,
and 192724 argand burners in shops, &c., lighted for
1000 hours? Ans. 95373^.
135. The gas consumed in London requires about
50,000 tons of coal per annum. For how long a time
would the gas this quantity may be supposed to pro
duce (at the rate of 9000 cubic feet per ton), keep one
argand light (consuming 4 cubic feet per hour) con
stantly burning } Ans. 12842 years and 170 days.
? * 136. It requires about 14,000 millions of silk worms
to produce the silk consumed in the United Kingdom
annually. Supposing that every pound requires 3500
worms, and that onefifth is wasted in throwing, how
many pounds of manufactured sill, may these worms
be supposed to produce ^ Ans. 1488 tons, 1 cwt., 3 qrs.,
17 1b.
137. If one fibre of silk will sustain 50 grains, how
many would be required to support 97 tb } Ans 13580.
• 138. One fibre of silk a mile long weighs but 12
grains ; how many miles would 4 millions of pounds,
annually consumed in England, reach }
Ans. 23333333331 miles.
139. A leaden shot of A\ inches in diameter weighs
17 lb ; but the size of a shot 4 inches in diameter, is to
that of one A\ inches in diameter, as 64000 : 91125 :
what is the weight of a leaden ball 4 inches in diameter >
Ans. 119396.
140. The sloth does not advance more than 100
How loRij would it
f o irn
to
IW!
1 f r
im
Dublin to Cork, allowing the distance to be 160 English
mil©8 ? Ans. 2816 days; or 8 years, nearly.
if
'i
i „ I
il ^:i\
202
COMPOUND PROPORT/ON.
141. li'ugliish race horsea l.^vn i. i
tl^oruto of 58 miles ai W t "^ '^ ^^ ^' "^
vclocify, „ii..i,^ *i,/ 1" 'loiii. In what time at n;.
iir'Xrt^ ^"^ll*'"'^ ^'* 3000 tons;
'^^or^^lZti tii r""" ^'rs' ^''"ut the
tlioii luto hairsmin J """i'^ny, mado into steel a,„l
wte, there aTo X3''Z"« /^'! ''''' "'"Jth"
gnuus of steel? ^,„ Sjooa™ ''°° "'""" ™"3
COMPOUND PEOPOKTfON.
^^rfp^^^^^^^^^^^^ «. although t.o"
i»'r."ft.t ratio „; th; thi.tt.erofl''''''"SH? '« «.e
. jr. I'ut down the term, nf ? 1 . P™P""'''»n
■"«.e first and second ace, if "V'" """^ ^••"™
antecedents may form one n!f ™'','' " ^V that the
mother In ^ttin^ Zu 7,7' "I "'" »™«eque„t:
oflect It has upon thf ansZif „?""' ""^M" "hat
'^'  ^w man, JJ^ ^:^^^^'^^^;;J« ^ a wall in 20
icily
i'^^^own{.32J,wiJlbea;a
Hows
m
COMPOUND PRoroirnoN.
'J03
^n to go at
^0, at til id
Cork bo
^000 tons ;
' of 5 loet
of 2 feet
about tliG
about ten
pound of
steel, and
Joductiiic/
t>ut 7000
Jgli two
stion, to
?>?• In
ios, one
to the
I.
ratios
at the
quents
what
idown
anto
rm as
livide
in 20
I
'3 •
ir • k \ '^o'^^'tiwis which givo 2U days.
20 days imperfect ratio.
1 days, the number sought.
17 men
37 yards
conditions which give the required number ol" Unys.
17 : 5 : : 20 : ?
10 : 37
And 17
10
5..oo.20x5x?>7
3y 17x10
'i'ho imperfect ratio consists of days — thoroforo we ar«i to
(lilt 20, the given number of days, in the tliird place. Two
ratios remain to be sot down — that of numbers of 7ncn, and
that of numbers of yards. Taking the former first, wo ask
ourselves how it affects the answer, and find tliat the more
men there are, the smaller the required numlierwill be— tsinee
the greater the number of men, the shorter the time ro{]uirod
to do the work. We, therefore, set down 17 as anlecedeut,
and 5 as consequent. Next, considering the ratio consisting
of yards, we find that the larger the number of yards, the
longer the time," before they are built — tlieroi'ore increasing
their number increases the quantity retiuired. Hence we
put 37 as consequent, and 10 as antecedent: and the whole
will be as follows : —
=130 days, n(;ariy.
45. The result obtained by the rule is the siinio .'is wonM lie
found by taking, in succession, the two j)roportioa8 supposed
by the question. Thus
. 1 5 men would build 16 yards in 20 days, iu how )uany
'8 woeld they build 37 yards '
.(3' : 87 : : 20 : ^" — number of days which 5 men would
16
require, to build 37 yards.
00 v37
If 5 men would build 87 yards in.tl_r2 — days, iu how many
16
days would 17 men build them ?
17 : 5 : : ?^ : 2^x517=20x5x37 ^^^ ^^^^^^
16 16 17X10
of days found by the rule.
40. ExAMPLK 2. — Tf 3 men in 4 days of 12 working hours
each build 37 perches, in liow many days of 6 working
hours ought 22 men to build 970 perches '.*
204
22
8
87
8 ::
12
970
COMPOUND PROPORTlOJf.
3 men.
4 (iixyH.
■12 hours.
37 porches.
? days.
8 Jiours.
22 mon.
970 porches.
" • " i'ui uiies.
3X12X97 0x4
22X8 X a/ """=21 i days, nearly.
days^U^islhtlmport't ?.r """''^fore 4
place The more moZhXl^^ S.o^5^ '' P"' '" "»« ^'"rd
form the work : therefore 29 jT * « *^''^*' necessary to per
smaller the nu^Cof worW E"* ^''?*' T*^ ^^ «««^""^I K
the number of days • hTnce 8^is Z fi *?' ^7' *^« ^'^^g''
The greater the number of perche?th^'*' ¥ ^^ «««""J
of days required to build tWn ^^'^ ^''^^^^ter the number
put first, and 970 second ' consequently 17 is to bo
or one in the first, a^ne in tL'"?! '^l? '^^"^ P^^«« J
same number. ' '"^ *^^ ^^^''^ place, by tho
Example 1 Tf ♦!
32 : IGO • • 8 • ^^^^X20x8 ^
5 : 20
32x5
Dividing 32 and IfiO ^« qo ,
1
1
5
4
8 : 5x4x8=100
measure a quantity in the S Z ''°^*?''' """^^^^ ^"'
place ; or one in the first on?' ^"l^°o*her in the second
This will in some iistanif 1 ^"°*^'' ^^ *^^^ ^^^'^ place
into unity^wSrrsrtr^^^^^^^^ ^'^ti:
COMPOUM) PROPOKriON.
205
irly.
therefore 4
n the third
^'^'•y to ncr
cond. The
the larger
12 second,
ho number
7 is to bo
hy divid
id place ;
J by tho
iles Costa
es cost ?
uotients.
pr{))or
!is long
)er will
second
' place
a^ntities
ExAMPLK 2— If 28 loads of Htono of IS^wt. each, build a
wall 20 foot lon^ and 7 foot hip;h, how nian\ loads of lU cwt.
wjll build one 323 feet long and 9 feet high ?
: 28 : 15x323x9x28 _^^^_
19
20
7
15 :
323
9
19X20X7
Dividing 7 and 28 by 7, we obtain 1 and 4.— Substitutine
those, we have °
19 : 15 : : 4 : 1
20 : 323
1:9
Dividing 20 and 15 by 5, tho quotients ore 4 aai 3 :
19 : 3 : : 4 : I
4 : 323
1 : 9
Dividing 4 and 4 by 4, the quotients are 1 and I :
19 : 3 : : 1 : ?
1 : 323
1 : 9
Dividing 19 and 323 by 19, tho quotients are 1 ,ind 17 :
1 : 3 :: 1 : 3x17x9=459.
, . 1 : 17
1:9
In this process we moroly divide the first and second, or
first and third terms, by the same number — which [29] does
not alter the proportion. Or we divide the numerator and
denominator of the fraction, found as the/oMr<A term, by the
Kame number— which [Sec. IV. 15] does not alter the quo
tient.
EXERCISES IN COMPOUNB PROPORTION.
1. If £240 in 16 months gains £64, how much will
d£60 gain m 6 months ? Ans. £6.
2. With how many pounds sterling could I gain
£5 per annum, if with £450 I gain £30 in 16 months ?
Ans. £100.
3. A merchant agrees with a carrier to bring 15 cwt
of goods 40 miles for 10 crowns. How much ought hi
to pay, in proportion, to have 6 owt. carried 32 miles )
Ans. IGs.
K 2
ii
at
U
ii i^.
:lf^«lf
P
■^f)(j
<o.ui.oi;nd phopoijtion.
Am. £20. ^'^'^^J " ^'"'"'''^ lUO iiiilos »
fo/q/^ f^^^ "' ''^ ^^^«»cIiandl.so aro carriod 40 ,niln«
for Js*^ itV/"^.^^'^""^^ "''^^''^ ^« rried 60 S«
fn. T 1^^^ * ^^ inorchandiso aro carried 9n ,«{i
1« honest™ Sr ie&AloO »r h' "'^* '" « ^'^^
would bo rcouimd f,^ 1 ''^ " ; """^ """V lioraM
days ? AnXof ^'"'^ """^ *'"' """^'"'^ '« 3
bein« i J toThtCS^^^^^^^^^^ wag.'
men's, and 24 paiv of won.onVIlmt' 1 P"'"' °^
each kind woiJd ic T 1 '''"'*'> V"" many pair of
13. A wall is r t,!' rl /■',"■ f women's shoes.
how'^JI/iJrri/d'S i^ZT "''"T' u
tunoei ,8 jays.^ .^„, X',^ tons the sa»o d. <
10. 11 ^/j,. are the wa.ws of 4 «,or, /•...  ^^
wages of
i^sn iOf
I
COMPOUND PROPORTIOM.
307
what Jill bo tlio wages of 14 iiiea for 10 day^P Am.
16. If 120 busliols of corn List U horses 50 davs
•7
• iu!^ ^^ '' ^?°^!''^" ^'"''^^^ ^^^ "»'^« i" 3 days when
the days arc 14 hours Jong, in how many day«ot'7 hours
each will ho travel 300 nriles ? Ans j^ " '^*' ^' ^ *'°"'«
• i^'o^/ the price of 10 oz. of bread, when the corn
,s 4.. 2d per bushel, be 5^., what wm.st'be paid ibr 3 b
12 oz when the eorn is 5s. r,d. per bushel ? ^1... 3 . 3/.
VJ. 5 compositors m 16 days of 14 hours lon^^ can
compose 20 sheets of 24 pages in each she , 50^11 .t"
days ot 7 hours long may 10 compositors compose a
Bltr filV'"'^^ ^"/^^^ """^ ^^^*«^' containi^ng 40
Bhects 16 pages in a sheet, 60 lines in a pae, and
50 letters m a line ? Avs. 32 days ° '
M^^p}^ ^'^' been calculated that a square degree (about
69X69 square miles) of water gives off by cvinor 
tion 33 millions of tons of water .er day. Ylow mu
laLVt 7'\ ^^Vr^P^'^'"^ ''' ''^'ole surface
to be 14 square feet ; and that the barometer stands at
31 inches ? Ans, 13 tons 19 cwt.
QUESTIONS IN RATIOS AND PUOPORTIOX.
1. What is the rule of proportion; and is it ever
called by any otlier name ? [IJ.
2 What is the difference between simple and com
pound proportion ? [30 and 421
3. What is a ratio ? [7].
4. What are the antecedent and consequent ? [71
o. VV hat IS an inverse ratio ? [8] .
6 What is tlie difffironno betwn"r» " ..:ii_.« 1
and a geometrical ratio ? [9j. ^^•^luat
1 r1
208
COMPOUND PROPORTION.
7. ftow can we know whether or not an arithmohV^l
or geometrical ratio, is altered in value p [10 and^' f'^
other? [Lj' "'"' '^''''''^'^ '^^'''''^ ^^ *^"^« «f ^"^
9 What is a proportion, or analogy ? [Ul
10. What are means, and extremes ? flSl
i.l%I^L:nir^^P''  Soometial .oan of
Jfc^roporL'rffe]''^' ^°« •!»««   arith
miLKri^T™lf=" f ■!««. a.e in gco
fo^M'^llyZii^'" P"P°'"""^' '^ three quantUie,
madt i!?T'"° the principal changes which may be
made^m a geometrical proportion, without destroying
r^li^;^"!? ^^ ?ecessarj, or even correct, to divide the
rule of three into the direct, and inverse .P [35]
18 How IS the question solved, when the first o,
second terms are not of the .same denomination • or one
19 hI^''^ ''^*^^'^ ?^^^^^"* denomination ? [371^
if I' ll^Z '' ^ ^""'^^''^ ^" *^^« rule of proportion solved
rf the thn^ term consists of more tha'n KeVoS
.J^f' ^^"""^ i^i*' solved, if fractions or mixed numbe^s
oTin^rth^tet3%ra» ^^ ^"^^^^
„f ^!' ^''° ^"^ °^ ""^ '''™« °f a question in the r„I„
1 1 'ii
f
209
A H I T JI M E T I C .
PART II.
SECTION yi.
. PRACTICE.
.t'»'SS jJis" '""'"•"s «.■ —.J
Mie latter '' ^ '^ *^'' ^'""'^" ^« *« "le price of
tlip niLoa '41 parts, and bndiaj' the sura of
« nart, " ,;"/n P"',"' •"■ ''y dividiug^tho price iZ
,>ot mJZTlZf ^^ a number, are those wf.ioh do
any TateJ^ /o'^^t^'^idt: VTo'l T"''"^'^ "^
Mwe have seen fSeo It 2fiV,L f^?"' P"'''' ""'
3 To find ti,t „r T J' „ "■* "'"t^'' measure it.
B „, . ^ • ? • T"' P"'^ of "ly number
ing Quofaflf.; \I ^l"?'' divisor, and the result.
fmtjU ; :,;! rtd^^ct „re;!::r:\':e ;7r
gte'n nurab:;:' "° ''^ ""'""""^ all «„t prrS'e " „'
810
PRACTICE.
parts^f 'ST""""^^^'"^ "'' *^'° ^^'""^' ^^^ «o«^Pound aliquot
2)84
2>!2 '
3)21
7)7
The prime aliquot parts aro 2, 3, and 7 ; and
2x2= 4^
2x3= 6
2x7=14
2x2x3=12 [ "^''^ "'° compound aliquot parts.
2x2x7=28
2x3x7=4'^
14:^21 *2ran'd42^'''^'' ^^""''^ '° '''^''' ^'' 2' ^' ^' ^' ^' ^2,
^ 5. Wo may apply this rule to appUcate numbers— Let it
2)240
2)jl20
2)G0
2)30
3)15
5)J5
2X2=
• 2x3=
2x5=
2x2x2=
2x2x3= 12=
2x2x5= 20= 1
2x3x5= 30= 2
2x2x2x2= 16= 1
2x2x2x3= 24= 2
2x2x2x5= 40= 3
2x2x3x5= 00= 5
2x2x2x2x3= 48= 4
2x2x2x2x5= 80= fi
2x2x2x3x5=120=10
4
G
10
8
d.
8
6
4
4
8
in shillings
PRACTICE.
211
And placed in order
£> d.
3
1
1!0
¥(5
I
4^?
' = 4
' = 5
tV= 16= 1
it
d.
4
1 8
24;= 2
V= 8
2^=12=1
d.
1= 30= 2 6
1= 40= 3 4
i= 48= 4
i= GO
5
A= 80= G
8
I i=120=10
Aliquot parts of a shilling, obtained in the same way
1 1
T? — T
_l I
2? 2
_l J
s. d.
Aliquot parts of avoirdupoise weight—
s. d*
4=6
Aliquot parts of a ton.
ton cwt. or
I ; 1 9
?o — 2 — ■^
20 ■■• ^
TIT— ^X= «5
tV= 2 = 8
J= 2.!=10
^= 4"=16
^= 5 =20
J=10 =40
Aliquot parts of a cwt.
cwt. K)
'=2
!
2?
■» = 7
8
T?
i=14
4=16
i=28
1=56
Aliquot parts of a quarter
qr. ib
A=2
1=14
, Aliquot parts may, in the same manner, be easily
obtamed by the pupil from the other tables of weights
and measures, page 3, &c. ^
6. To find the price of a quantity of one dcnomina
tion— the price of a " higher" being given
Rule.— Divide the price by thai number which ex
presses how many times we must take the lower to
make the amount equal to one of the higher d'enomina
pei^^cwrr'~^^''* '' *^^ P''"' ^^ ^"^ ^^ ^^ ^"**^^ ^t '2..
Tl^^f'''"!*^^^^^ ^\^^ ^ «*""^ « ti"^es, to make 1 cwt.
Therefore the price of 1 cwt. divided by 8, or 72s ^SoI
IS the price of 14 ib. ^ , ux <^. . o—y*.,
The table of tiliquot pn'^^ of avoirdimms,> ^xo^rht hr^s,
ti;f pile'' o'F l*ti"'' " ^' "• ^■'■'•f""'" F i*^ '!>; J rf^
lis
ill
212
PRACTICE.
1.
2.
3.
4.
EXERCISES.
What is the price of
i cwt. at 29s. 6d. per cwt. ? Ans. Is. 4^d
, a yard of cloth, at Ss. Gd. per yard ? An!. 4s. 3d
14 ft, of sugar at 45.. 6d. p.r cwt. ? Ans. 5s. S^d
What IS the price of ^ cwt., at 50.. per cwt. >
£ s. d.
505.=2 10
qra. cwt.
The price of 2=i is
of 1=1^.2
IS
]
£> s.
0=2 10^2
12 G=l 542
5
. Therefore the price of 2f 1 qrs.(= cwt.) is 1 i7g
J« half the price of 2 qrl'' Th&.rth " pi^'e'of I'llTu
onf ow" ^"'' '' ' '^' P^"^ *^ ^^^ «f '^"f tlfe ^ile of
6.
6.
7.
8.
. What is the price of
? oz of cloves, at 9.. 4rf. per lb ? Am. 3id
nail of lace, at 15.. 4d. per yard ? Ans."" '
2 10, at 23s. 4d. per cwt. .? yl?^.?. ii^
^ ib, at 18.. Sd. per cwt. ? Ans li^."
14^/.
7. When the pric3 of wwrc ^A^w o^j^ 'qow>r'' dono
minatiou is required— ^^^
Ia«f vnf;""^'^^^^' P""' °^ '^^^^^ denomination by the
last rule ; and the sum of the prices obtained will be
the requued quantity. [ '^^
at «;.;•:? ct'"?^' " "" P™= °f 2 <!■ 1* » of gar,
s. d.
.45 price of 1 cwt.
cwt AnAoo A ti ^ . [or ,1 of 1 cwt.
14\b=f,or'of2or8 ^■^' ^^ ^•^••^/=22.. G,i.44, is the
s, or , 01 ^ qrs. p^ce of 14 lb, the i of 1 cwt..
\ ^^i^~T; . 01^ the { of 2 qrs.
2nr« t n . i H IS the price of 2 qrs. 14 lb.
^ ^J^«rT5 ?f 1 cwt. Therefore 45.^ (f,li« r^vj^. of 1 o.f ^  o
or zos. Qu., 18 the price of 2 qrs, ^' ' o^,t.J~2,
PRACTICE.
213
Ans. 4s. 3d
Ins. OS. S^^d
r Gwt. ?
d.
£ s.
0=2 10^2
_G=1 5h2
G
\vt. : md ita
nd its price
of f cwt. is
lie i^rice of
. S^d.
ns. llid.
3r" dcno
on by tlie
d will be
of sugar,
of 1 cwt.
Bof2qrs.,
^4, is the
of 1 cwt.,
14 lb.
nTt.)f2,
45^^ JoJ f, ^ rV ^i^.y^^i^f 2 qrs. Therefore
T rJo ? ; f?' S'f7'^=5* 7Arf., is the price of 14 !b.
f; mT. • l"^^'^^'^'.^.' *^^ P'^^*'^ ^f 2 qrs. plus the price
ot 14 lb, IS the price of 2 qrs. 14 lb. i i f
EXERCISES.
What is the price of
9. 1 qr., 14 fe at 46^. 6d. per cwt. ? Ans. 17s. 5id.
10. 3 qrs. 2 nails, at 17*. 6c?. per yard ^ Ans
los. 3r/.
11.5 roods 14 perches at 3s. lOd. per acre ? Ans.
5s. l^d.
12. 16 dwt. 14 grs., at £4 4s. 9d. per oz. ? Ans
£3 10s. 3}d. ^
13. 14 lb 5 oz., at 25.5. 4d. per cwt. ? Ans. 3s. 2fd
8. When the price of ow, "higher" denomination is
required —
KuLE. — Find whaf ninnbor of times the lower deno
mination must be taken, to make a quantity equal to
one of the given denomination ; and multiply the price
by that number. (This is the reverse of the rule eiven
above [G]). ^
ExAMPLK.— What is the price of 2 tons of sucar, at 50s.
per cwt. <
1 \?.^'^'^' ^^ *^® ^'« ^^ 2 tone : hence tlie price of 2 tons will
be 40 times th price of 1 cwt.— or 50,9.x40=£100.
50.S. the price of 1 cwt. multiplied
^y 40 the number of hundreds in 2 tons,
gives 2000,s.
or XlOO as the price of 40 cwt., or 2 tons.
EXERCISES.
What is the price of
14. 47 cwt., at l.y. S^. per lb I Ans. £438 VSs. 4d
15. 36 yards, at 4d. per nail r Ans. £[) V2s.
16. 14 acres, at 5s. per porch .? Ans. £5f)0.
17. 12 R), at lf/. per grain ? Ans. £504.
IS. 19 hhds., at 3d. per gallon : Ans. i214 19*. 3^.
0. When the price of more Ihitu one "higher" dcno
miuatiou is required —
ill
214
PRACTICE
• RuLE.—Find the price of each bv the lasf nn^ nrU
atlTr"o"^eT^* " *'^ ^'^^ <^^ ^ «*• ^ <1 of flour,
1 stone is the j\ of 2 cwt. Therefore
tv,„u r ^ 1. ,?^''*^o price of one stone, '
multiphed by B, the number of stones in 2 cwt,
gives 3a?., ;• ^?ce ot 16 stones, or 2 cwt.
EXERCISES.
in e , What is the price of
.£1 L ^ ' ^ ^''•' ^ °^"'' ^* ¥' P«^ ^^« ^ ^^^.
20. 6 cwt. 14 ib at 3^. per ib ? Ans. £8 Us. 6d.
^1. 3 ib 5 oz. at 2id. per oz. ? Ans. 9s. lUd.
ul^fieMTlfc.' ^^^^«>3P^^> at 5. per perch .P
fini^/i. ^^^"^ *^.^ P"^^ °^ ^"^ denomination is given to
find the price of any number of another— ^ '
±tuLE..Find the price of one of that other denomi
nation, and multiply it by the given number of the
^^XAMPLE.What is the price of 13 stones at 255. per
1 stone= cwt. Therefore
8)25^, t he price of 1 cwt. divided by 8,
we obtain £2 7^ M the price of 13 stones.
Istoneistliejoflowt. Hence 25.!.^8=3s 1 u i, th.
pnoe of one stone; and 3,. lirf.xlS, the price of is '.it!
PRACTICE. 816
EXERCISES.
What is the price of
24. 19 lb, at 2d. per oz. .? Atis. £,2 \0s. Sd.
25. 13 oz., at \s. Ad. per ft) ? Am. \s. \d.
26. 14 ft), at 2s. Qd. per dwt. ? Ans. &420.
27. 15 acres, at 185. per perch ? Ans. ^£2160.
28. 8 yards, at Ad. per nail ? Ans. £2 2s. Sd.
29. 12 hhds., at 5d. per pint ? Ans. ^£126.
30. 3 quarts, at 91^. per hhd. ? Ans. Is. Id.
11. When the price of a given denomination is the
aliquot part of a shilling, to find the price of any num
ber of that denomination —
Rule. — Divide the amount of the given denomina
tion by the number expressing what aliquot part the
given price is of a shilling, and the quotient will be the
required price in shillings, &c.
Example. — What is the price of 831 articles at 4d. per }
3)831
277s.=£13 17s., is the required price.
Ad. is the  of a shilling. Hence the price at Ad. is i of
what it would be at Is. per article. But the price at Is. per
article would be 831s.:— therefore the price at Ad, is 831s. ^ 3
or 277s.
lii
EXERCISES.
What is the price of
31. 379 ft) of sugar, at 6d. per lb ? Aiis. £9 9s. 6d.
32. 5014 yards of calico, at 3d. per yard.? Ans.
£62 13s. 6d.
33. 258 yards of tape, at 2d. per yard ? Ans. £2 3s.
12. When the price of a given denomination is the
aliquot part of a pound, to find the price of any number
of that denomination —
Rule. — Divide the quantity whose price Is sought
by that number which expresses what aliquot part the
given price is of a pound. The quotient will be the re
quired price in pounds, &c.
216
PRACTICE.
^ J«Mr,..._What is tI.o price of 1732 ft of to., .t 6,
would bo jeiT^p fi. r P°.^ ^^ ^^^^ at ^1 per lb it
EXERCISES.
Wliat is tho price of
35. 13 oz., at 4.. per oz. ? Ans. £2 12s.
37. 83 a, at Is. Ad. per ib ? Ans. £5 10s Sd
39* 976 rV'f o'^ P°^ ^^ ^ ^ ^3 16.. si.
4?' "? »»' ^* ^^>^ pel ft ? ^.^..^2 6.. 8^
44. 1000 ib, at 35. Ad. per lb .? ^w.. jgieg 13,. 4J.
13. The complement of the Drice i«i wliof ;f ,„„ * i.
pound or a shilling ^ ^* ^* ^^^*^ ^^ »
por^."d r"^''"* '•'"'» Prf^o "f 1«0 yard., at 13», «.
Cs. Sd. (the complement of 13s. 4^) ia 1 of £1
Z" f 'S *■>» Pri"* »t *1 per yard
.uhtraot ^, the ^rioo at fe. sS. (tCoomplement)
andthedi£fe™oo, 980, will ho the price at ^^Spor yard.
I47^™tTlf\t]^:.1;:;' t'J^O at C. 8. are e,„al to
price of 1470 at^lSs, «34hepr1oe^ofl?70 at £?'"'•*'''
.the price of 1470 at Gs. &(. per yard ' '°""
PRACTICE.
217
..■i
tea, at 5j
2 lb is tht,
1 per lb it
> 135. 4d.
12 7s. 6d.
Sd.
8d.
3 9*. 2d.
•
13*. 4d.
tnts of a
uot part
lot —
ho caso
IcuJated
13s. 4cl
1.
lement)
3r yard.
qual to
ice the
I minus
EXERCtSEiS.
What is tho price of
45. 51 ib, at 175. 6^. per lb .? Ans. £44 125. 6a
46. 39 oz., at 7d. per oz. ? Ans. £1 2s. 9d.
47. 91 ft), at lOd. per ib ? Ans. £3 15s. lOd.
48. 432 cwt., at 165. per cwt. } Ans. £345 12s.
14. When neither the price nor its complement is
the aliquot part or parts of a pound or shilling —
Rule 1. — Divide the price into pounds (if there are
any), and aliquot parts of a pound or shilling; then
find the price at each of these (by preceding rules) : —
the sum of the prices will be what is required.
Example.— What is tho price of 822 lb, at £5 19s. SJtZ.
per lb I £5 m. ^d.=£6+m. Zld.
s. d. £>
8 —I
!fM
But 195. ZM.
<
10
G
2
0=1
or jV of the last
or ^ of the last
Henco tho price at £5 195. Z'^d. is equal to
£
822x5
82a
£ s. d.
=4110 0, tho price at
822
3
8 22
It
)l22?^123_i_ A\
= 411
= 274
= 102 15
= 5 2 9
= 17 14
£ s. d.
5 per lb.
£i or 10
£l or 6 8
£] or 2 6
;; £ I ov 1J
;;4i;oro ooi
11
11
And X4903 14 lOi is the price at £5 19 3 J „
The price at tho whole, is evidently equal to the sura of
the prices at each of tho parts.
If the price were £5 19s. 3^d. per lb, we should sub
tract, and not add the price at }d. per lb ; and wc then
would have £4902 05. l^d. as the answer.
15. Rule 2.— Find tho price at a pound, a shilling,
a penny and a farthing ; then multiply each by thoir
[ ,
II
f '!
t
iJI
n
m^:^'
218
PRACTICE.
produl'' Tf""^' m' '" '^' Sivon piico ; and add the
pioaucta. Using the same example
£ ». d. £ a
^llui S S5j;»®P^!«eat£l)x 5=4110 6
Its Q X (*?0P^'ceatl«.)xi9= 780 18
^ 1? J.Sf'oP'Jceatle/OX 3= 10 5
i7 li(tho price at 4flf.)x 8= 2 11
d.
the price at £5
» 19*.
6 » 8d
4i „ id.
And the price at £5 Ids. 3rf. is ^£4903 14 lOi
tlie^hi.w''/'"^^^. *^' P^'^°" ^' *»^« "«^t number of
the highest denomination ; and deduct the price at tlia
difference between the assumed and given prFce
Usmg stiU the same example—
prfco.'' '''''' *' ^^'^^ ^'^^''^ denomination in the givea
From the price at £6 ^ "' ^' nr 4?qo n n
Deduct the pri^e( the price at 8rf.=27' 8 > ^ ^ ^
"'*'''' < .. 4rf.= IZlijO' 28 5 U
at 8id.
The diflFerence will bo the price at £6 19*. 8 or i^lTm
17 Rule 4.— Find the price at the next hijrher
atT <^ff ' '^ ' Tf^^ '' ^""^"g ' ^d deduct the p1:Se
at the difference between the assumed, and given price
ExAMPLE.What is the price of 84 lb, at Qs. per tb^
ds.^Gs. 8d. minus 8J.=^ minus i^10.
we have ^25 4 0, the price at 6s.
EXERCISES.
^r. ^o « ^^^^'^^ ^^ *^^ P^ice of
49. 73 ib, at 13^. per ft ? Ans. £^7 9s.
i>0. 97 cwt., at 15^. Qd. per cwt. ? Ans. £1Q 7s.
51. 43 ft, at 3s. 2d. per ft ? Ans. £6 'l6s. 2d
9d.
r)2. 13
OS. lid.
53. 27 yards, at 7*. ' 5U.
Is. llid.
acres, at £4 5s. lid. per acre.? Am. £55
per yard.? Ans. JBIO
llsTtl t' p™' '^ "" "™ °""»''^' °f ^'i»S».
PRACTICE.
219
add tlio
>rioe at £5
„ Ids.
8d
id.
imber of
30 at tlid
lie given
a.
i2
d
!8 5 li
3 14 lOi
higher
he price
n price
pep lb.
7.1. 9d.
d.
. £10
illings,
Rule. — Multiply the number of articles by half the
number of Hhillings ; and consider the tens of the pro
duct as pounds, and the units doubled^ as Hhillings.
Example. — What i{ the price of G4G lb, at 16s. per lb 1
646
8
510
£510 10,s.
2s. being the tenth of a pound, there are, in the price,
half as many tenths as shillings. Therefore half the number
of shillings, multiplied by the number of articles, will express
the number of tenths of a pound in the price of the entire.
The tens of these tenths will be the number of pounds ; and
the units (bcigg tenths of a pound) will be half the required
number of shillings— or, multiplied by 2— the required num
ber of shillings.
In the example, 10.9., or £8, is the price of each article.
Therefore, since there are 040 articles, 040xi^8=ii5.108
is the price of them. But 8 tenths of a pound (the unitHn
the product obtained) , are twice as many shillings ; and henco
we are to multiply the units in the product by 2.
EXERCISES.
What is the price of
54. 3215 ells, at 6s. per ell.? Ans. £964 lO.v.
55. 7563 lb, at 8s. per lb } Ans. £3025 4..
56. 269 cwt., at 16s. per cwt. } A"'!. £215' 4s.
57. 27 oz., at 4s. per oz. .? Ans. £ 8s.
58. 84 gallons, at 14s. per gallon ." Ans. £58 I6s.
19. When the price is an odd number of shillings,
and less than 20 —
Ruj^E. — Find the amount at the next lower even
number of shillings ; and add the price at one shilling.
Example.— What is the price of 275 lb, at 17s. per lb ?
8
220
13 15
The price at 16s. (by the last rule) is
The price at Is. is 275s.=
Hence the price at 16s.+ls , or 17s., is £233 15s.
i
Ik
Ih
220
PnACTICB
Tho prico at 17.v. is equal to tlio
prioe at oiio aJjiliing.
price at IGs., plus the
EXERCISES.
59. 86 oz., at 5*. per oz. ? Ans. £'21 10.?
60. 62 cwt at 195. por cwt. ? Ans. £i)S ' ISs.
ro ^^yf^^'^^ 175. per yard.? yl^i^. X'll is*.
02. 439 tons, at 11^. per ton ? Ans. jC;241 9.v.
Od. 96 gallons, at 7s. per gallon ? Ans. i233 12^.
number'''' *^° ^""""^'^^ '' ^^P^^^^^^^^'l l>y a mixed
liuLE.— Find the prico of tho integral part. Then
t7o"n t^d d ^'ny^'' ^ « nume?ato/of the fr J.
ti^on and divide the product by its dcnoniinatortho
quotien will be the price of tho fractional part. To
sum of these prices will be the price of the whole quan
^ ExAMPLE.What is the price of ^ lb of tea, at 5.. per
The price of8 lb is 8x5?.=
The prico of  lb is ^^'•
£ f. (I
2
3
9
And the price ofS^ lb is . . 2~T~U
of a
EXERCISES.
64.
65.
66.
67.
4id.
68.
4s. ed.
69.
iP2751
What is the price of
oIqIT""' ?* ^^ ^f' P*^^ ^°2«" ^ ^^^•'. 175. 101./.
xi/dy It), at 25. brf. per lb .? ^^5. ^£34 3^ ] j^"
5302 lb, at 145. pei: lb ? Ans. 371 IO5. 'e/ '
178f cwt., at 175. per cwt. ? Ans. i2151 125
7023 cwt., at £1 12s. 6d. per cwt. .? Ans. £1239
^\^^\.,?*' ^^ ^^ ^' '^^' per cwt..? Ar^.
lis. b^d.
PRACTICK.
221
s.f plus the
lS,s.
18*.
9.V.
!3 12s.
a njixcd
t. Then
tho frac
%tOT — tho
rt. Tho
Die quan
at 5.'?. i)cr
)v\co of a
s. 101(1.
'. Ud.
6(L
51 125
^1239
' Ans.
21. The rules for finding tho pnco of Bovoral deno
minations, that of one being given [7 and 9], may bo
abbreviated by those which follow —
Avoinliopoise Weight. — Given the price per cwt., to
find the price of hundreds, quarters, &c. —
lluLE. — Having brought tho tons, if any, to cwt.,
multiply 1 by tho number of hundreds, and consider tho
product as pounds sterling ; 5 by the number of quar
ters, and consider tho product as shillings ; 2^, tho
number of pounds, and consider the product as pence : —
tho sum of all the products will be tho price at £1 per
cwt. From this find tho price, at the given. number of
poun '''=', shillings, &c. ,
Example.— What is the price of 472 cwt., 3 qrs., 10 lb,
at £5 \)s. 6(1. per cwt. 1
£ s. d.
I 5 2'
Multipliers 472 3 16^
472 17 10 is the price at £1 per cwt.
5
2589 1 9 the price, at £5 9s
At £1 per cwt., there will be £1 for every cwt. We mul
tiply the qrs. by 5, for shillings ; because, if one cwt. costs £1,
the fourth of 1 cwt., or one quarter, will cost the lourth ot
a pound, or 5s.— and there will be as many times 5s. as there
are quarters. The pounds are multiplied by 2\ ; because it
the quarter costs 5s., the 28th part of a quarter, or 1 lb,
must cost the 28th part of 5s., or S^c/.— and there will be as
many times ^d. as there are pounds.
EXERCISES.
What is the price of
70. 499 cwt., 3 qrs., 25 fib, at 25s. lid. per cwt. ?
Ans. £647 I7s. l\d.
71. 106 cwt., 3 qrs., 14 ft), at 18*. M, per owt. ?
Ans. jeiOO 35. 10rf.
PRACTICE,
^I'^mo 55'.'4.r ' '^' =" '''■ '"■' P«' '• ?
jB4?i7l° eZ*'' ^ '■'''"■' '* '''' *' '*'• "''• P""' •='"■ ■' ^'"•
jes^ss^sij''' ^ '^^^'' '' "^' "' ''™ "''• P" ™' • ^'"•
A«:£L r iwf/"' '" *' ^' '^' '"■ p=' '• •'
^riri 7^/ '■■■' '' ''' "* '''■ ^ p°' '••'
22. To find the price of cwfc., qrs., &c., the price of
a pound being given —
RuLE.—Haviug reduced the tons, if any, to cwt.,
multiply 9. Ad. by the number of pence contained in
the price of one pound :— this will be the price of one
"""^i?"!. I^^ *!'^ P"^"' °^°"^ ''^^ ^'y 4, and the quotient
Will be the price of one quarter, &c.
Multiply the price of 1 cwt. by the number of cwt. •
the price of a quarter by the number of quarters ; the
price of a pound by the number of pounds ; and the sum
o± the products will be the price of tlie given quantity.
Example.
8(/. per lb. ?
d.
What is the price of 4 cwt., 3 qrs., 7 lb, at
9 4
8
s. d.
28 R « f ^ •'' ''H '"^^ Xj.^i" give 298 8 the price Of 4 cwt.
28)18 8 e price of Iqr. X3,willgive 66 the price of 3 qrs.
8 the price of 1 lb X7, will gi ve 4 8 the price of 7 lb.
20)369 4
And the price of the whole will be £17 l£~i
At Id. per lb the price of 1 cwt. would be l\M. or 9i. Ad. •—
therefore the price por cwt. will be as many times 0*. U as
there are pence in the price of .a nnnnd. The >iri" p 
quarter IS \ the price ot 1 cwt. ; an<l there will bo as many
times the price of a quarter, as there are quarters, &c.
PRACTICE.
223
!l»'ifl
EXERCISES.
What is tlie price of
79. 1 cwt., at 6d. per ft) > Am. £2 16s.
80. 3 cwt., 2 qrs., 5 ib, at 4d. per Bb .? Atis. £6
12s. 4d.
81. 61 cwt., 3 qrs., 21 lb, at 9d. per Bb } Am. £2\B
2s. M.
82. 42 cwt., qrs., 5 ft), at 2bd. per lb r iln*. ^2490
105. bd.
83. 10 cwt., 3 qrs., 27 Bb, at bid. per ft) ? Am.
£2Q\ Us. Qd.
23. Given the price of a pound, to find that of a ton —
EuLE. — ^Multiply £Q 6s. 8d: by the number of pence
contained in the price of a pound.
Example. — ^What is the price of a ton, at 7d, per ft) '?
JS s. d.
9 6 8
7
65
6 8 is the price of 1 ton.
If one pound cost Id., a ton will cost 2240rf., or £9 6s. Sd.
Hence there will be as many times £9 6s. Sd. in the price
of a ton, as there are pence in the price of a pound.
EXERCISES.
What is the price of
84. 1 ton, at 3^. per ft) > Am. £28.
85. 1 ton, at 9d. per ft) .? Am. £84.
86. 1 ton, at 10c?. per ft) ? Am. £93 6^. 8d.
87. 1 ton, at 4d. per ft) T Am. £37 65. Sd.
The price of any number of tons will be found, if we mul
tiply the price of 1 ton by that ntmiber.
24. Troy Weight. — Given the price of an ounce — ^to
find that of ounces, pennyweights, &c. —
Rule. — Having reduced the pounds, if any, to ounces,
set down the ounces as pounds sterling ; the dwt. as
shillmgs ; and the grs. as halfpence : — this will give the
price at £1 per ounce. Take the same part, or parts,
&c., of this, as the price per ounce is of a pound.
224
PRACTICE.
Example 1 What is the price of 538 oz., 18 dwt, 14
grs., at lis. 6d. per oz. ? » , ^^ uwt , i^
Us. Gd. =ilL.^J4.ij^2
£> s. d.
2)538 18 7 is the price, at ^1 per ounce.
^ol^fr 1 « i?f ^' *^® P^^^«' a* 10^ per ounce. ^'
^ 1 Q n li }^ *^® P"^®' «* !«• per ounce.
•^^ '^ 5f 18 the price, at 6d. per ounce.
And 309 17 8i is the price, at lb. 6d. per ounce.
14 halfpence are set down as 7 pence.
If one ounce or 20 dwt. cost £1, 1 dwt. or the 20th part
^4th part of 1 dwt., or 1 gr. will cost the 24th part of
IS. — or ^o. ^
^E^xAMPLE 2.What is the price of 8 oz. 20 grs., at .^£3
£> s. d.
8 10 is the price, at £1 per ounce.
o
Price It £1 ' 100 ifi ? • t5^ P"^®' ""^ 5^ P^^ ^^''e
vTnl I o A n . 1. i^ *^e P"«e, at 2s. per ounce.
I nee at 2s.^ 4 =0 4 0^ is the price, at 6d. per ounce.
And £25 2 7^ is the price, at ^£3 2s. 6(i. per oz.
EXERCISES.
What is the price of
88. 147 oz., 14 dwt., 14 grs., at 75. 6d. per oz. .?
4715. £55 7*. 11^^. p "^..
, 89. 194 oz., 13 dwt., 16 grs., at 11*. 6d. per oz. ?
Ans. ieill 18*. 10^(£. ^
90. 214 oz., 14 dwt., 16 grs., at 12*. 6d. per oz. }
Ans. £134 45. 2d. ^
91. 11 ft), 10 oz., 10 dwt., 20 grs., at 105. per oz. ?
Ans. £71 55. 5eZ. "^
92. 19 ft), 4oz.,3grs.,at£2 55. 2(^. peroz. .? ^W5.
£523 185. lli<^. ■^
93. 3 oz., 5 dwt., 12 grs., at £1 65. 8<^. per oz. ?
Ans. £4 7*. 3J</.
llHi
' ^i
PRACTICK.
225
18 dwt, 14
ice.
ice.
ce.
je.
' ounce.
3 20th part
. ; and tlio
th paxt of
5rs., at .£3
r ounce.
per ounce,
per ounce*
per ounce.
Qd. per oz.
per oz. ?
per oz. ?
per oz. }
per oz. ?
? Ans.
per oz. ?
25. Cloth Measure. — Griven the price per yard — to
find the price of yards, quarters, &c. —
Rule. — Multiply £1 by the number of yards ; 55. by
the number of quarters ; Is. 3d, by the number of nails ;
and add those together for the price of the quantity at
£1 per yard ? Take the same part, or parts, &c., of this,
as the price is of iSl .
Example 1.— What is the price of 97 yards, 3 qrs., 3
nails, at 8s, per yard ?
£1 5s. Is. Zd.
MuItlpUers 97 3 2
2 )97 17 6 is the price, at £1 per yard.
5)48 18 9 is the price, at IO5. per yard.
From this subtract 9 15 9 the price, at 2s. per yard.
And the remainder 39 3 is the price, at 8s, (10s.— 2s.)
If a yard costs £1, a quarter of a yard must cost 5s. ; and
a nail, or the 4th of a yard, will cost the 4th part of 5s. or
Is. 3fZ.
Example 2.— What is the price of 17 yards, 3 qrs., 2
nails, at £2 5s. 9d. per yard ?
£1 5s. Is. Zd.
Multipliers 17 3 2
17 17 6 is the price, at £1 per yard
35 15 is the price, at £2 per yard.
The price at £l^ 4=4 9 4i is the price, at 5s.
Th© price at 5s. ^ 10=0 8 11 is the price, at 6d.
The price at GcZ.f 2=0 4 5 is the price, at 3^.
And £40 17 9^ is the price, at £2 5s. 9d.
EXERCISES.
What is the price of
94. 176 yards, 2 qrs., 2 nails, a 15s. per yard > Ans.
jei32 9s. 4^d.
95. 37 yards, 3 qrs., at o^l 5s. per yard ? Ans, £47
3s. 9d.
96. 49 yards, 3 qrs., 2 nails, at £1 10s. per yard.?
A.ns. £n 16s. 3d.
97. 98 yards, 3 qrs., 1 nail, at £1 15s. per yard.?
Ans. £172 18s. d\d.
H
226
PRACTICE.
98. 3 yards, 1 qr., at 17s. 6d. per yard.? Am £2
16s. lOid.
99. 4 yards, 2 q^" , 3 nails, at ^1 2s. Ad. per yard ?
Am. £5 4s. S^d.
26. Land Pleasure. — Kule. — Multiply £1 by tLe
number of acres ; 55. by the number of roods ; and l^d.
by the number of perches : — the sum of the products will
be the price at £1 per acre. From this find tho price,
at the given sum.
Example. — What is the rent of 7 acres, 3 roods, 16
perches, at J£3 8s. per acre '?
£ s. d.
Multipliers
15 1^
7 3 16
Sum of the products 7 17 0, or the price at JEl per acre.
23 11 the price at £Z per acre.
3 18 6 the price at 10s. per acre.
27 9 6 the price at £3 10s. per aero.
Subtract 15 8i the price at 2s. per acre.
And 26 13
Si
,j is the price at £3 8s.
If one acre costs £1, a quarter of an acre, or one rood, must
cost 5s. ; and the 40th part of a quarter, or one perch, must
cost the 40th part of 5s. — or Ikd.
•'I n
EXERCISES.
What is the rent of
100. 176 acres, 2 roods, 17 perches, at £5 6s. per
acre ? Ans. iE936 Os. 3d.
101. 256 acres, 3 roods, 16 perches, at £6 6s. 6d.
per acre .? Ans. ig 1624 lis. 6id.
102. 144 acres, 1 rood, 14 perches, at £5 6s. 8d. per
acre ? Ans. £769 16s
103. 344 acres, 3 rbods, 15 perches, at £4 Is. Id.
per acre > Ans. £1398 Is. Id.
27. Wine Pleasure. — To find the price of a hogs
head, when the price of a quart is given —
Rule. — For each hogshead, reckon as many pounds,
and shillings as there arc pence per quart.
PRACTICE.
227
Am £2
per yard ?
1 by the
and l^d.
)ducts will
tho price,
roods, 16
per aore.
r acre.
iv acre.
t. per aero.
V acre.
.8s.
rood, must
erch, must
15 6^. per
'6 6s. 6d.
s. 8d. per
4 Is. Id.
F a hogs
y pounds,
Lkkvivve. — What is the price of a hogshead at dd. per
quart? A)is. £9 ds. .
One hogshead at Id. per quart would bo 63X4, since there
are 4 quarts in one gallon, and G3 gallons in one hhd. But
G;>x4d.=252<i.=<£l Is.; and, therefore, the price, at dd. per
quart, will bo nine times as much — or 9X^1 ls.=£0 9«.
EXERCISES.
AVhat is the price of
104. 1 hhd. at 18^. per quart ? Ans. £18 185.
105. 1 hhd. at 19<^. per quart? Ans. £19 195.
106. 1 hhd. at 20d. per quart ? Ans. £21.
107. 1 hhd. at 2s. per quart ? Ans. £25 45.
108. 1 hhd. at 25. 6d. per quart? Aiis. £31 105.
When the price of a pint is given, of coxirse we know that
of a quarc.
28. Uiven the price of a quart, to find that of a tun—
Rule. — Take 4 times as many pounds, and 4 times
as many shillings as there are pence per quart.
Example. — What is the price of a tun at lid. per quart *
£ s.
11 11
4
4G 4 is the price of a tun.
Since a tun contains 4 hogsheads, its price must he 4 tiinp,*
the price of a hhd. : that is, 4 times as many pounds and shil
lings, as pence per quart [27].
EXERCISES.
What is the price of
109. 1 tun, at VJd. per quart? Ans. £79 IGs.
110. 1 tun, at 2Qd. per quart ? Ans. £84.
111. 1 tun, at 25. per quart ? Ans. £100 I65.
112. 1 tun, at 25. 6d. per quart? Ans. £126.
113. 1 tun, at 25 8d. per quart? Ans. £134 85.
29. A nmnher of Articles. — Given the price of 1
article in p(nice, to find that of any number —
KuLE. — Divide the number b.y 12, for shillings and
■1 I
11!
I<
I i
228
PRACTICE.
pence; and multiply the quotient by the number of
ponce m the price.
ExAMPLE.WIiat is llio price of 438 articles, at 7d caol^
12)438
365. Gd, the price at Id. each.
20 )255"~6
£12 15 6 the price at 7(/. each.
438 articles at 1<Z. each will cost 438<;.=36s. Gt^. At 7d each
2i¥l5" 6^ '' ''' muchor 7X3G.. Gd.JMs.'SuJ:
EXERCISES.
What is the price of
114. 176 ib, at 3d. per lb .? Ans. £2 4s.
115. 146 yards, at 9d. per yard ? Ans. £5 9s. 6d
] ;^ ]^^ y^^^,^' '^* 101^. per yard ? Ans. £7 Us. 64
117. 192 yards, at 7id. per yard ? Ans. £6.
118. 240 yards, at 8^d. per yard ? Ans. £S 10s
30. Wages.—mymg the wages per day, to find
their amount per year — ° r j>
RuLE.Take so many pounds, half pounds, and 5
pennies sterling, as there are pence per day.
Example.— What are the yearly wages, at 5d. per day ?
1 10 5
5 the number of pence per day.
7 12 ] the wages per year.
£l^?nfS P^I'^''y;^« equal to 365rf.=240./.+120./.+r,J.=
*iflU* +6^/. Therefore any number of pence ner dav mimt
be equal to £1 10.. 5d. multiplied by that number ^*
What is the amount pei year, at
119. 3d. per day.? Ans. £4 Us. 3d.
120. Id. per day.? Ans. iglO 12*. 11^.
121. 9d. per day.? Ans. iE13 13*. 9^.
122. 14^. per day.? Ans. £2\ 5*. IQd.
123. 25. 6d. per day .? yl^w. ^'41 I*. 3^.
124 ^d. per day .? Am. j(212 ISs. (]\d.
pnACTfc:E.
229
BILLS OF PARCELS.
Mr. John Day
Dublin, IQth April, 1844.
BouKlit of Richard Jones.
(/.
15 yards of lino broadcloth, at 13 per yard
24 yanU of tsuperHtu) ditto, at 18 9
!27 yards of yard ^vido ditto, at 8 4
il) yards (;f drugg(it. at .
1 2 yards of sorp;*', at
?)'2 yards of shalloon, at .
S
2 10
1 8
£ s.
10 2
22 10
11 5
5
1 14
2 1?>
d.
()
4
Ans. jCSS 4 10
Mr. James Paul,
pair of worsted stockings, at 4
(J pair of silk ditto, at . . ir)
17 pair of tliread dllto, at . 5
2'.) pair of cotton ditto, at . 4
14 pair of yarn ditto, at . 2
15 pair of Women's silk gloves, at 4
I'J yards of Ihinnel, at . . 1
Dublin, mMa\j,\UA.
Bought of Thomas Norton,
s. d.
G per pair
„
4 „
10 „
4
2
1}, per yard
A,^s. £23 15 4J
Mr. James Gorman,
40 ells of dowlas, at
;') 1 ells of diaper, at
31 ells of Holland, at
2'.> yards of Irish cloth, at
\~\ yards of muslin, at
13'j' yards of cambric, at
s.
1
1
5
10
Dublin, nth May, 1844.
Bought of John Walsh & Co
G per ell
4^ „
8 „
4 per yard
2\
(i
[>'t yardsofpi'iMtod calico, at 1 2}
5J
11
Ans.
£h4 5 10]
2'30
l'n.\(TlVK
mi
■I h
Liuly Denny,
.'jjimlaofsilk, at . . 12 9 per yard
]■> yards of floAvered do., at 15 G
11^yardsof Instring, at . 10
J 4 yards of brccado, at . 11 3
12] yards of satin, at . 10 H
llg yard,i of velvet, at . 18
Dublin, 20th Maij, 1841.
Bought of Iviuhard Mercer
Mr. Jona^i Darling,
5;
15' lb of currants, at
1 7 lb of IMalaga raisins, at
l^i* lb of raisins of the sun, at
17 lb of riee, at
SI ib of j)Opper, at . . . x k>
;> loa\ OS ol'sugar, weight 32,1 lb. at 81
i:.'. oz. of cloves, at / . " '. D"per"oz
Ans. £U 15 10
JDublin, 2lst May, 1844.
Bought of William Roper.
•s\ d.
4 per lb
0"
'6i
1 6
n
Aus. £3 13 OJ
• Mr. Thomas VVnght, """'"' '' "* ^""'' ^'^^^
Bought of Stephen Brown & Co.
252 gallons of prime wliiskey, at 4 per gallon
2r)2 gallons of old malt, at .08
252gallonsof old malt, at . 8
!)
Ans. i:204 12
MISCELLANEOUS EXEUCrSES.
What is the price of
i4 ISa". '2^u.
2. a.>4 lb, at ]\d. per ib .? Ans. £1 ](]s. IQid
Jl. 47o6 lb of sugar, at I2}d. per ib .^ Ans. £242
ins. Id.
.10
£127 \Qs.
25 pair of silk stoc
kings, at C>s. per pair ? A
'tis
rUACJ'UK.
231
Ans.
i.
£242
Am
5. 3751 pair of gloves, at 2^. Cxi. ? Aois. £469 55
6. 3520 pair of gloves, at 3a. Gd. ? Ans £(516.
7. 7341 cwt., at £2 C)s. per cwt. ? Ans. £16884 Gs.
8. 435 cwt. at £2 7s. per cwt. f Ans. £1022 5s.
9. 4514 cwt., at £2 lis. l},d. per cwt. } Ans.
£13005 lO.v. ?,d.
10. 3749f cwt., at £3 15a. ChI. per cwt. } Am.
£14153 17i. iJJr/.
11. 17 cwt., 1 (jr., 17 lb, at £1 4i. 9^Z. per cwt. .?
£21 lOi'. Sid.
12. 78 cwt., 3 qr.s., 12 lb, at £2 17.?. M. per cwt. .?
Ans. £227 lis.
13. 5 oz., G dwt., 17 grs., at 5^. lOrZ. per oz. ? Ans
£1 11. S. 11,/.
14. 4 yards, 2 (rs., 3 nails, at £l 2.?. 4d. per yard .'
A:ns. £5 4s. S^d.
15. 32 acres, 1 rood, 14 perches, at £1 16^. per
acre .^ Ans. £5S 4.9. If^Z.
16. 3 ir:dlons, 5 pints, at 7a'. 6^/. per gallon.' Ans,
£1 7,v. 21 d.
17. 20 tons, 19 cwt., 3 qrs., 27} ft), at £10 10^
per ton .? Aius. £220 \)s. 1 1^^/. nearly.
18. 219 toas, iij cwt., 3 qrs., at £11 75. Gd. per
ton r Ans. £2500 13a. O^r/.
QUESTIOXS IN PRACTICE.
1. Wluit is practice .' [I].
2. Vriiy is it so called .' [1].
3. What is the dilforeneo between ali(uot, and aliqnant
parts.' [2J.
4. Ihm are the ali(Uot parts of abstract, and of
applieato mniibers found } [3].
5. AVHiat is the difFoiencc between prime, and coni
pouiid aliipiot parts t [3j.
fi. [fow is the pri(!e of any denomination fnmd, tli;it
of another beins given ? [(i and .8].
7. llrtn" is the prho of two or more denoi:iination3
found, that of one bein;z uivin ? [7 and 9j.
S. The p;icG of ou:.> dc^nominatirM! being given, how
do wo find that of any number of another . [ lOj.
I
r H
232
PAACTICiC
r. ^; F''?.,v '^ ^!''''" ^^ ^''y tlonomination is the aliquot
part ot a shilling, how in the price of any number of that
tlononuimtion louud ? [llj.
10 AVhen the price of any denoniiuation is the
ahquot part of a pound, how is the price of any num
ber ot that denomination found ? [12].
ri3 I ' ^^^'^' ^^ ^"^^"^ ^^ ^^'"^ complement of the price r
li. When the complement of the price of any deno
minaum ih the nli^uot part of a pound or shilling,
but the price IS not so, how is the price of any number
ot that denomination found ? [13 1.
13. When neither the price of a given denomination,
1 or Its complement, is the aliquot part of a pound or
M.iUmg, how do we find the price of any number of
that denomian'ion ? [14, 15, lb', and 17] .
14 How do we find the price of any number of
articles whon t.^e pri(,c of each is an even or odd num
ber ot slullrngs, and less than 20 ? [IS and 19].
lo How is tlio price of a (pumtity, represented by a
Tiaxod number, found ? [20] . ^
1<; How do we find the price of cwt., qrs., and lb,
when the price of 1 cwt. is iven .' [21].
17. How do we fin.l the price of cwt., qrs., and ib,
when th.^pvieoof 1 11, isgiven.^ [22].
18. h\ vi' is the price of a ton found, when the prico
of 1 In IS given ? [23]. ^
10. How do wo hud the price of oz., dwt., and grs.
wh?.i the price 5f an ounce is given } [24]. '
^20. How do we find the price of yards, qrs., and nails,
wucii tlie price of a yard is given .' [25].
21 How do we lind the price of acre^, roods, and
porches .' [PH].
22. How may the price of a hhd. or a tun be found,
wlion the T;,ri'jo of a <piart is givcm ^ [27 and 28].
^ 23 Hov; rnny tlie price of any number of articles bo'
round, the price of each in pence bciut^ given ? [29]
. ^^^" ,^i*''\'^''« ^^'=^K«« per year found, those  ■ .• day beino
given ■' [30] ^ ^ *
TARE AND TRET.
233
»ffp m
■15^
TARE AND TRET.
3'. The ^jrross \vei<^ht is tlio wei;;lit both of the
goo('h, uii«,l oitho bag, &c., in which thoy arc.
Tare Ih an jiHowanco for the bag, &c., wliich contains
th(! aiti('h\
Suftle is the wci«;ht which renmins, after deducting
the tare.
Tret is, uynally, an allowaneo of 4 Vb in every 104 ft),
or j'jj of the weight of goods liable to waste, after the
U\V{>. has been d^'dueted.
Cioff is an allowance of 2 To in every 3 cwt., after
both tare and tret \m\ti been deducted.
What reuKiius after makin!? all deductions is called
the vd, or nc/il wciu;ht.
Diffjrent allowances are made in dilTercnt places,
and for different goods ; but the mode of procetMlin;^ is
in ^v^l ca^ii'S very simple, and may be understood from
the ibllowin<ic —
EXERCISES.
1, I>ouo'ht 100 carcasses of beef at 18;$. ikl. per cwt.;
gross wciglit 450 cwt., 2 qrs., 23 lb ; tret 8 lb per car
cass. What is to be paid for them ?
100 carcasses.
8 lb per carcass
■ — cwt. qrs. H:)
Tret, on the entire, 800 ib=7 IG
cwt. qvc. lb.
Cross 450 2 23
'J "ret 7 10
443
2 7 at 18s, (Jd. per cwt.=.£410 5s. lOJfL
2. What is the price of 400 raw hides, at 195. 10//.
per cwt. ; tlie f^ross weight bein;^ 300 cwt., 3 q'.'s., l.o
11) ; and the tret 4 lb per hide ? Ans. £290 3.7. 2^d.^
3. If 1 cwt. of butter cost £3, wh:it will be the price
of 250 firkins; gro.na weight 127 cwt., 2 qrs., 21 tb ;
tare 11 lb per firkhi .? Aiis. £:^(}i) Ss. 0id. ^
4. ^Vhat is the price of 8 cwt., 3 (rs., 11 lb, at Ins. \
fuL per cwt., allowing the usual tret .'^ A its. JLJti 1 Ia\
lO^i/.
234
TAKR AND TRET.
G. What U tho price of 8 cwt. 21 lb at lfi» 4i./
por cw . , a I.nvin. the visual fret ? ^L ^7 4/ 81./
(>. Jou.h ^.^ h .J,s. of tallow ; No. 1 wcighin. ] J tt
qr, 11 11, tare 3 <r.s, ^u lb ; and No. 2, II cwt nr '
t.u>;';rtect^;t;rtV^^^^ ^vL^^S;;
A. . . cwt. qrs. lb.
Orosa Meight of No. I, K) 1 n
(iioss weight of No. 2, 11 17
(JrosH weight, .
iuie,
Snttlo,
Tret 1 lb per cwt.
cwt
. 21
. 1
2
3
. 1!)
.
2 22
19^§
cwt. qrs. lb.
i are 3 20
TuroO 3 14
r~3 G
, i. X^5 "tI}'!/. '' ^ '^ The price, at 30,. po,
prupo^ti::^'' thirthef tret n.ay be ibund by the following
cwt. cwt. qrs. tb.
1 : I'J 2 22
1
lb.
1().'10
7. What is tho price of 4 hhds. of coppnra.s • No 1
WCM,.,,, gross 10 cwt..,2,p..,4 tb,tafo^^;'s. 4lh?
1^ cwt 1 rp., tare 3 qrs. I4 lb ; No. 4, 11 cwt 2
'I's M I,, tare 3 q.s. IS ib ; the' tret b in" 1 7b ' r
<;t.,nKl the price 10. per cwt. .^ An! mIoiM.
«. Wliat will 2 bags of merchandise corao to • No 1
w,Mg]in,g gross 2 cwt., 3 qrs., 10 lb No o 'q .J'
and at l.v. br/. per lb .? yj^^. £59 2* 8V
^. A merchant has sold 3 bags of pepper • No 1
Weighmo mo.SS 3 nwt 9M.a . V. °o . i'\IW5 ^^^Ji
No. 3,
1 It
cwt. 2 qrs. ; No. 2, 4
do th
.> cwt., 3 qrs., 21 lb; tare 40 1b
•per cwt. ; and the price boin'^ lor/
cwt., 1 qr., 7 lb
y come to ? Ans. £74 1.?. 73.3^
por bag; tret
per lb. What
1
10. liought 3 packs of wool
<F., 12 lb; JN'o. 2,3
weighing. No. 1 , 3 cwt.
'■? qis., ].) ib ; tare 30 lb
cwt., 3 qrs., 7 Ib; No. 3, 3 cwt
"'> Rtoue; and at 10.?. 3^
per pack ; tret 8 lb f(
amount to ^
per stone. Wliat do tl
or f'v.'Mv
 ■ J
Je»
iAui: ANij via: I.
835
Vo. 1,
No. 2,
No. 3,
owt. qrs. lb.
3 1 12
3 3 7
3 2 15
tb.
Tare 30
'J'aro 30
Taro 30
IJross, 10 3 6
'I'aro, 3 6
St.
1
90—3 qrs. G lb.
Suttlo, 10 0=70 stones.
st. st. lb. lb.
20 : 70 :: 8 : 28 =
st. lb.
Suttlo, 70
Tret, 1 12
lb.
J2
Not weight, 08 4, at lO.*. Gd. por stono=<£35 16s. 7}^d.
1 1 . Sold 4 packs of wool at 9^. ^Jd. por stono ; woi<^h
ing, No. 1, 3 cwt., 3 qrs., 27 lb. ; No. 2, 3 cwt., 2 qrs.,
10 lb. ; No. 3, 4 cwt., 1 qv., 10 lb. ; No. 4, 4 cwt.,
jjr., G lb : tato 30 lb per pack, ami trot 8 It) for every
20 stono. What is the price > Ans. £49 lbs. 2^^%d.
12. Bought 6 packs of wool ; weighing, No. 1, 4 cwt.,
2 qrs., 15 lb ; No. 2, 4 cwt., 2 qrs. ; No. 3, 3 cwt.,
3 qrs., 21 lb ; No. 4, 3 cwt., 3 qrs., 14 lb ; No. 5, 4
cwt., qr., 14 lb : tare 28 lb per pack ; tret K lb for
every 20 stone ; and at lis. 6d. per stone. What in
the price ? Ans. £11 15s. Sfjfc?.
13. Sold 3 packs of wool ; weighing gross, No. 1, 3
cwt., 1 qr., 27 ft) ; No. 2, 3 cwt., 2 qrs., 16 lb ; No. 3,
4 cwt., qr., 21 lb : tare 29 lb per pack ; tret 8 lb for
every 20 stone ; and at lis. Id. per stone. What is the
price } Ans. £4\ 13s. l\\\d.
14. Bought 50 casks of butter, weighing gross, 202
cwt., 3 qrs., 14 lb ; tare 20 lb per cwt. What is the
net weight }
cwt. qrs. lb.
Gross weight, 202 3 14
Tare, . .30
qrs. cwt.
O I
iZ?
14 = i
cwt.
202
qrs.
o
tt).
14
20
4040 lb.
10
5
1
o
T
2
of
of
25
Net weight, 166 2 161
i of the last, \ =the tare on 3 qr. 14 lb.
2i = Xof thelast,
Tare, 4057^ lb = 30 cwt., qr., 251 lb.
I'
236
TARE AND TRET,
lo Ihc gross weight of ton liluls. of tallow is 104
cwt Z qrs., 2b lb ; and the tare 14 lb per cwt. WHU
IS the net weight ? Ans. 91 cwt., 2 qrs., 141 lb
«J^"/^^ gioss weight of six butts of currants is 58
ewt., 1 qr., 8 lb ; and the tare 16 lb per cwt. What is
the net weiglit .? Ans. 50 cwt., qr., 1^ ft,
17. What is the net weight of 39 cwt!, 3 qrs., 21 lb •
the tare benig 18 lb pe. cwt. ; the tret 4 lb for 101 lb •
and the cloff 2 \\, for every 3 cwt. > " ^^ ^^ l il> ,
cwt. qrs. lb.
(xross weight, 39 3 21
Tare,
lb. lb. cwt.
18= ! 1^='
^^ ^ 2=1^8
cwt. qrs. lb.
39 3 21
5 2 23
2 24
1 13
Suttle, .
Tret=2'gtli, or
Tare, 6 1 13
2 lb m 3 cwt. is the .^,th part of 3 cwt. i, u
Hence the cloff of 32 cwt. 26 lb is its ^^,th part, or
33 2
1 1
32 2<
2
4
O >
Net weight, 32 4
18._ What is the net weiglit of 45 hhds. of toliaeco •
weighing gross 224 cwt., 3 qrs., 20 lb ; tare 25 cwt'
3 qrs. ; tret 4 lb per 101 ; cIoiT2 lb for every 3 cwt ?
Am. 190 cwt., 1 qr., 14^^ lb. ^
19 What is the net "weight of 7 hhds. of sn^nr
weighing gross, 47 cwt., 2 qrs., 4 lb ; tare in the whol..'
10 cwt., 2 qrs., 14 lb ; and tret 4 ib per 104 \h > An^
3o cwt., 1 qr., 27 lb. ^ ' '
20. In 17 cwt., qr.. 17 lb, gross weight of V^alls
how much net ; allowing 18 lb per cwt. ta?e ; 4 lb per'
104 lb tret ; and 2 \h per 3 cwt. cloff? Ans. 13 cwt.,
6 qrs., 1 lb nearly. '
QUESTIONS.
1. What is the gross weioht ? fSll
2. What is tare? [31]. ^ ^ "''
3. What is suttle .? [31 ] .
4 What is tret.? [311 '
5. What is cloff.? [31].
6. "Wliat is the iitt weight.? [31].
7. Are the allowances made, always the same ? [31],
qrs
lb.
3
21
1
13
o
i>
Ji
1
4
20
O >
Am.
237
. i;
SECTION VII.
INTEREST, &c.
1. Interest is the price which is allowed for the nso of
money ; it depends on the plenty or scarcity of the latter,
and the risk which is run in lending it.
Interest is either simple or compound. It is simf^h
when the interest due is not added to the sum lent, ^^,
as to bear interest.
It is compound when, after certain periods, it; is made
to bear interest— being added to the sum, and considered
as a part of it.
The money lent is called the principal. The sura
allowed for each hundred pounds " per annum" (for a
year) is called the " rate per cent." — (per iilOO.) The
amount ia the sum of the principal and the interest due.
SIMPLE INTEREST.
2. To find the interest, at any rate per cent., on any
Bum, for one year —
lluLE I. — Multiply the sum by the rate per cent.,
and divide the product by 100.
Example.— What is the interest of £072 14s. U. for cno
year, at 6 per cent. (XG fur every £100.)
£> s. (J.
672 14 3
4030
20
5 G
725 The quotient, £40 7s. 3f/., is the iLtereet required.
'J.Ort
J lii
We h
ave divided by 100, by merely altering the decimal
point [Se<3. I. 34J
O'
kS
IN CERKST.
II till Interest w'.'i'c 1 percent,., it woulrl be the Imiidredth
part of the principal— or the pjineip;il multiplied l)y ^^f, ; but
being b per cent., it is t> times iid much— or tlie principal mul
tiplied by y^.
3. Rule II. — Divide the interest into parts of cClOO;
and take corresponding parts of the principal.
EsAMPLK.— What is the interest of £32 4s. 2d., at G per
cent. '. ^
£G = £5+£l,ov£^^lAns£^^^5. Therefore tho in
terest is the J^ of the principal, plus the I of the J.
£
20)32
s. d.
4 2
5) 1 12 2r} is the interest, at 5 per cent.
6 5] is the interest, at I per cent.
And 1 18 7f is the interest, at 6 (5+1) per cent.
'* EXERCISES.
1. What is the interest of ^£344 lis. Qd. for one year,
at per cent. > Ans. £20 1 3*. 1 0\d.
2. What is the interest of .£600 for one year, at 5 per
cent. ? Am. i230.
3. What is the interest of dE480 15.y. for one year, at
7 per cent, t Jlns. ii33 135. 0rZ.
4. What is the interest of ^£240 10s. for one year, at
4 per cent. > Am. £9 12s. Md.
_ 4. To find the interest when the rate per cent, con
sists of more than one denomination —
^ lluLE. — Find the interest at the higliest denomina
tion ; and take parts of tliis, for those which are lower.
The sum of the results will be the interest, at the given
rate.
Example. —What is the interest of £97 8s. 4d., f(y one
year, at £o lO.s. per annum '?
£5 = £yi,0; and 10.'. = £,5,.
£ s. d. '^
20)97 8 4
10)4 17 5 is tho interest, at 5 per cent.
9 9 is the interest, at lOs. per cent.
And 5 7 2 is the interest, at £5{10s. per cent.
£5
INTEKEST.
239
At 5 per cent, tho intevcst is the .1, of fhe piMneipal ; at
lOs. per c.nt. it is the j\ of wluit it U at 5 .er cent. There
fore, at £5 lOi'. per cent., it is the sum of huth.
5. What is tho interest of /J371 IDs. ly. for one
year, at i:'3 155. per cent. ? Am. jSIS 18.v."llf^.
ei. AVhat is the interest of i^84 ll.y. lOirZ. for one
year, at £4 5.v. per cent. } Ans. j£;3 II.9. lOfr/.
7. What is the interest of JCOI O5. 3<Z. for one year,
at £6 12.9. 9^/. per cent. } Am. M O5. \0\d.
8. What is the interest of £ms bs. for one year, at
£b 14a. i)d. per cent. } Am. £bb 8a. 8r/.
5. To find the interest of any sum, for several
years —
Eui.E. — Multiply the interest of one year by the num
ber of years.
Example.— AVJiat is the interest of £32 145. 2(/. for 7
years, at 5 per cent. '?
£ A. d.
20)32_14_2_
i 12 81^ is the interest for one year, at 5 per cent.
And 1.1 S 11^ is the interest for 7 years, at 5 per cent.
This rule requires no explanation.
EXEilCISKS.
9. Vvniat is the interest of £U 2s. for 3 years, at (3
per cent, r Ans. £2 U)s. dd.
10. What is the interest of ,£72 for 13 years, at m
10a. per cent. } Ans. £m IG.v. 9*^/.
11. What is tho interest of £Sb3 Qs. i)\d. for 11
years, at £4: V2s. per cent. } Ans. £431 vSs. l^d.
().. To find the interest of a given sum for years,
niontlis, &c. —
lluu:. — Having found the interest for the years, as
aheady directed [2, &c.], take parts of the interest
"jr that of tho mouths, &(
the results.
and tl'sn add
<U
! I
'■rt.
l^ui
I j,jj i
Hi
^•40
INTEREST
Example —What is the iz^tercs,* of J£86 85. M. for 7 vcars
and 5 months, at 5 per cont. > ^
20)86 8 4
4 G 5 13 the interest b> : T?a>, ai 5 jxv • cep
1 « o.~^"" A S *if ?^ *^'® interest for 4 monthv
1 8 9^^4 =0 7 21 IS the interest for 1 month.
And 32 llj is the required interest.
EXERCISES.
12. What is the interest of ^£211 5^. for 1 year and
6 months, at 6 per cent. .? yl^w. ^19 0.9. 3d
13 AVhat is the interest of £514 for 1 year and 7i
months, at 8 per cent. > Ans. £66 16^ 4U '
11. What is the interest of £1090 for l' year and 5
months, at 6 per cont. } Ans. £92 I3s
1;^ What is the interest of £175 lO.^. 6^. for 1 year
and J monriis, at 6 per cent. .? Ans. £16 135. 5//^.^
o. A\Iuit IS the interest of £571 15.. for 4 years
and 8 months, at 6 per cent. .? Ans. £160 1.. 9^./
17 VViiat IS the interest of £500 for 2 years and 10
iiiwntlis, at 7 per cent. ? Ans. £99 3.y. Ad
lb". What is the interest of £93 17.':. Ad. for 7 yc^as
,}^T'^^'f' ^'^' ^ P^^' «^"t • ^'I'i'^. ^14 11.. lid "
and ^ ^ ;f '•' /'' ^"^''''^ '^^ ^'^^ '^^ ^'^^ f^^ 8 y^^^
and S mouths, at o per cent. > Ans. £36 ll.v. 111,/.
O) or b, &c., per cent. '
At 5 per cent. — *
■ KuLE.— Consider the years as sh:lHn..s, and the
montlis as pence ; and find what alipot pa^ or part
^fS^^^n^r^^"^ ^^*^^^4^tor^a..
To find the interest at 6 per cent., find the interest
at 5 per cent., mid to it add its fifth part, &c.
Ihe mtcrest at 4 per cent, will bo the
per cent minus its fifth part, &c.
interest at
INTKRE6T.
2^1
d. for 7 years
4^ 5 p<\ • cep
s.
bh.
1 year and
ear and 7^^
yx^ar and 5
for 1 year
''• ^1 0*^
3r 4 years
s. 9^(1,
ars and 10
■or 7 years
Is. l\d.
n 8 years
v. 111^/.
y time, at
and tliG
t or parts
t or parts
LJ interest
iterest a^
^ 8. Example L— What is the interest of X427 5^. 9c/. for
years jind 4 luuiiths, at 5 per cent. '\
years and 4 months are represented by 6s. 4r/. : but
0.. 4,/.=...s..+],.+W.==<}J^ of a pou)td + the i of ths J,.
4)427_ .5__0_
5)100 To ,5y is tlie I of principal.
;])21 7 31 is the ^V(l of]) of principal.
_!._r_A ''^ ^^'"^ s'" <^'^ ''^' ^'') ^''" Pi'i'^^ip'd.
And l;>.5 G 1^ is the required interest.
The intcres^t of £1 for 1 year, at 5 per cent., would be 1..
lor 1 inouth 1</. ; for any number of years, the same nuinl)or
ot shillnigs; for any number of nioiitlis, the same number of
pence ; and for years and months, a corresponding nund>er of
shiUmgs and pence. But whatever part, or parts, these sliil
xHigs and i>ence are of a pound, the interest of any other sum
tor tJie sauio time and rate, must be the same part or parts of
tJiat otlier sumsince the interest of any sum is proportional
to the interest of JSI.
KxAMPiE 2.— AVhat is the interest of £14 2s. 2d. for G
years and 8 months, at G per cent. ?
G.'v. 8^/. is the .' of a pound.
£ s. d.'
3)14_2^2_
5)4 14 {)} is tlio interest, at 5 per cent.
18 W'l is the interest, at 1 per cent.
5 12 KJi is the interest, at G (5fl) per cent.
EXERCISES.
20. Find the interest of .£1090 17.?. Qd. for 1 year
and 8 months, at 5 per cent. .? Am. £90 18.v. \\d.
21. Find the interest of £976 14,?. Id. for 2%^ears
and 6 months, at 5 per cent. } Am. £122 \s. 9rZ.
22. Find the interest of £780 17s. fi^i. for 3 years
and 4 iiiontlis, ut G per cent. > Am. £1.^)6 'M. 6d.
23. What is tlie interest of £197 lis. for 2 years
and 6 montli.s, at 5 per cent. ? Am\ £24 13.?. l{)ld.
24. What is the interest of £279 lis. for 74 months,
at 4 per cent. . A as. £6 19s. 9,^,//.
7U(
Jit;
2.1. What IS iho. iuterest of £790 IGs. for 6
and S months, at 5 per cent. > A'us. £263 12j.
year
^m
y
212
INTKREST,
26 What is the interest of ^^121 2s. \U. for 3 years
and .3 inontiis, at 5 per cent. ? Ans. £oq 3^ 53,^ "^
27 ^VliMt i,s the interest of i2l837 4*. 2d. foVs'vca.a
and K) inontlis, at 8 per cent. } Am. £563 S.y. 3d.
9. When the rnU^ or nuin])er of years, or loth of
thein, are expressed by a mixed nuinber
l.ULE.— Find the interest for 1 year, at 1 per cent ,
and multiply this by the number of pounds and the frac
tjun of a pound (if there is one) per cent. ; the Mini uf
these products, or one of them, if there is but one, will
give the interest for one year. Multiply this bv' the
number of years, and by the fraction of i year (if^lhe e
f one) ;_ and the suni of tliese products, or one of them'
If there is but one, will be the required interest
atfpoTce'atT*"^ *'' interest of i:21 2.. 0./. fur 3 years
£21 2s. G./.^100=4.. 2ld. Therefore
.t s. d.
4 25 is the interest fori year, at 1 per cent.
lit Jo.
1 1 1' is the interest fori year, at 5 per cent.
a 1 ? 1 A n^ the interest for 3 years, at do.
i> lo luj IS the interest for 2 ofa year (£1 Is. l^^^.x"'),;
3 19 31 is the interest for 3^ years, at do.
J^^':.^''^ '' *^i'^terestof £300fbr5.years,
£300^100=3 LUhe interest fori year, at 1 per cent.
9 OLs the interest fori year, at 3 per cont.
__ ___^^ tlie interest for 1 year, at £■' (cC3x^)
11 5 is the interest fori year, at 3 per cent.
56 5 Is the interest for 5 years, at ^ por cent
 10 3 IS tlio do. for l pru (i:5 12^. G^r/.J2A
And 04 13 9 is the interest I)
r^■■t
or ir^ years, at Z} do.
'or 3 ycarfl
, 6^1.
fur 3 ycai'3
<s. 3d.
•1* both oC
por cent.,
(1 the fruc
he sum vi'
t ono, Avill
lis by the
' (if (lierc
B of them,
!t.
>r 3 years
x;'),utJo.
5^ years,
per cent,
oer cont.
por cent.
' peroenl
t 3^ do.
rXTERRST.
EXRRCISKS.
243
28. What is tliG interest of JE379 2s. Gd. for 41 years,
at 5 1 por cent. ? Aiis. £<ji 5.5. 5^/, ^ ^ »
29. What is the interest of .£640 \Qs. Qd for 24
years, at^4 per cent. > Am. £72 1,?. 2j\d.
30. "What is the interest of £600 IO.9. Qd. for 3i
years, at 5 per cent. } Ans. £11.5 2^. QJfZ. ''
31. What is the interest of £21? ^s^.\id for e^
years, at 5 5 per cent. .? Ans. £81 8,?. bld. "*
10. To find the interest for days^ at 5 per cent —
'/^.^^•— ^^"Itiply the principafby the number of days,
and divide the product by 7300.
E.VAMPLK.— Whatis the interst of i;2G 4.s. 2d. for 8 days?
£> s. d.
2G 4 2
8
201) 13 4
20
4193
12
r300)50320(6:!'W.
•43800 '
6520
U ItVZT'"'] "l^T'*/.' .^'^« ^' ^' 7^— «ince tlie remainder
IS gieatey than lialf the divisor.
The interest of £1 for 1 year is £J^, and for 1 day Jjf3G5=
20K3tl5="300; that is, the 7300th part of the principal.
Therefore the iritorest of auy other sum for one day, is the
.?00th part of that sum; and for any number of da^s. it a
that number, multiplie.l by the TSOOtli* part of tl>e princ ipal
nunrbpl'of T "'' 'T^ '^'/"f' '^'' ^'^""^^^^ multiplied bf he
number of days, and divided by 7300. ^
EXKRCISES.
33. Find the interest of £140 lO.f. for 76 days, at 5
per cent. Ans. £1 9s. 3^%^jd.
33. Find the interest of £300 for 91 days, at 5 per
cent. '•'■'• ^"^ '' no., 7 ^ ' 1^^
Ans. £3 I4s. Q^d
34. What is the interest of £800 fur 61 day
i ; i
,i'
, ^ 1
per cent. ? Am. £6 1.3.s. S%%d
s, at b
241
I.NTKKKST.
11. To fmr] t],o intovost f<n dnys, at .^;/7/ other rate—
pitrls of llii
KXAMI'MO
fliiys, at cCG
X3324 Gs.
£ .V.
5)5
2)1
OJU
And G lo"
je 1+105.
This rule
1 111,1 the mtercj.st at 5 per cent., and take
■* tor the remainder.
—What is the interest of Xr.324 C.<. 2<Z. for 11
ivs. per cent. ?
2^.Xll^7;500=£5 0.. 2],/. Therefore
2 is the interest for 11 days, at 5 per cent.
J .J IS the interest for 11 days, at 1 per cent.
JJ_ IS the interest for 11 days, at 10,v. per cent.
2J is the interest for 11 days, at jGG 10*. (£5f
requires no explanation.
EXERCISRS.
w ?f' ^u'^V" *^»e interest of £200 from the 7th May
to .the 2bth September, at 8 per cent. } Ans. £Q As
30. mat is the interest of ^£150 15^. Qd. for 53
aays, at 7 per cent. } Am. £\ \0s. 7^d
37. What is the interest of d8371 for l"year and 213
days, at 6 per cent. .? Ans. i235 os. Qd
A.!ll' .Tl''''^ "' ^\ ''l^'^'f^ °^ ^'^^ ^'''' ^ y^^'' and 135
aays, at 7 per cent. } Ans. £23 0.s\ 3^^;?.
Sometimes the number of days is"the\liquot part of
a year ; m which case the process is rendered more easy
l^^^^:ti^^ '^''''' '' ^^'^ ' ' ^ ^
1 year and 73 days=] i year. Hence tlie required intercsit
il75 '^r7'' ^'" 1 /f +i.t« lifth part. ]^ut Ihe interest':?
fjiio tor 1 year, at the given rate is £14. Tlierefore its
£1g'ig1 gi^«^ti>^« i« X14+i:y=:£i4+i2Ta..i
12. To find the interest for mon/Jis, at 6 per cent^
±IULE.— if the number expressing tlie months is even,
nmtoply the prmcipal by half i/>f number of month
and dn,de by 100 But if it is odd, multiply by tho
hali of om Jess than the number of montlis ; divide the
result by 100 i and add to the quotient what will bo
obtained if we divide it by one les than the nn.mIo of
11
INTEUIiST
345
! ti
KxAMiT.K 1.— What is the interest of X72 (Js. id. for 8
months, iit G per cent?
£ ,s. (I.
72 4
4
£2H0 5 4
20
1785s. The required interest h £2 Us. lOUl.
12
1024r/.
4
090= J f/. nearly.
Solving the question Ity tiic rule of three, wo shall have —
;eiOO : i:72 Gs. 4^/. : : JCG : £72 6.s. 4f/.x8xG
12 : 8 l(Xrxl2 =(^^ivul.
ing Itoth numerator and denoniinator by G [Sec. IV. '
X72G.9. 4r/.x8xG^G .£72 Gs. 4r/.xS
100 x 12^0 = lUO X 2 ~" = (dividing ooth
numerator and donominator by 2) ' ' " ■^^^yczo_J_o'~~
£12 G.;. Aily^A
100"
— that is, the required intore.sfc is equal to tlic given sum,
nuiKiplied by half the number which expresses the montliH,
and divided "by 100.
ExAMPi.E 2.— AVhat is the interest of £84 Gs. 2d. for 11
months, at G per cent. ? 11=10+1 10^2=5.
£ s. d.
84 6 2
•) One loss than the given number of
niHol^ inonths=10.
20
£ s. d.
4oOs. i0)4 4 Ojj is the interest for 10 niontliii, at U ptT oc'iit.
12 8 5 is the intore!.t lor 1 nujutl), ut Siinie rattj.
370(7. And 4 12 9 is the iutciest lor IJ (IDjI) moutUi, ut ti ;•*
4
280f.=.^J. nearly.
Tlie interest
11 1 month, pUis the "udovost of II — I monlh r1 1 — 1
11 months is evidently the interest ivf
I
.M
948
INTEItnsT.
ual >
'IXKKCISES.
39 What k tlio intorost of £250 17s 6d for <(
«fr ,;,,.'"■ "!,"""■"■'"'''' ■^■5^1 15s. for 8 months.
«*'. per (,..(. ^«. Jtaa 17s. 4.id '
6 pt or„t.".' "i!;.: i^'^rres"' ^'''° '"' ' "■"'"■'^' ^'
6 p1fr ofntt ^f i"r^' "' ■'"'' '" '" '™"*^' "'
at P,i^ "^^^r^^^^t ''■ ' ' ""■»'
...i' v^ir^*^ *''? '"''"'™' '« ?»■<! l>y ^nj/s, multinlv the
sam by the nnmboi of days which have elamcd b^fom
any payment wa. made. ' Subtract Z trjfavu „,
and multiply the remainder by the number of S
wh.eh passed between the first and second paymon?,
Ir r ^"'" "'T'^ P^>'™""' »■«' '""'"ply «l.is nS:
seconJ !nd "ir^'' "^ ''"^^ '^'™'' passL/ between the
days more £20 : in 15 mor^ £^o "^ i f .f'*'"^^ ~^ '^ • ^" '^
day,, an, the ^SeTat"' Cf Jlrj^^e^'ireit
X days. £ day.
nZx 6= 702xi 1
lOOx 7= 700x1
i^0xl5=1200xlf=°C^^^<^
48x60=3168x1 J
Wo?e''''* '^'^"' ^^^ 1 ^y *S Ver cent., is 15.. 9c/
INTERRST.
24/
..... i
fi)0 15 \)'^ is tho iutorost, at 5 pei cent.
'^ 2^ is tho interest, at I per cent.
Jl)0 18 11^' i8 tho intorest, at per cont.
G 4 is tho interest, at 2 per cent.
And 1 5 3J is tho interest, at 8 per cent., for th*^ given
sums and iiiuos.
If tho entire sum were G days unpaid, tho interest would be
tho aafoe as that of 6 times as much, fbr 1 day. Next, £100
due for 7 days, sliuuM proiluce as much as £700, for 1 day,
&c. And all tlio sums du ■ for tho diffcrci>t periods should
produce as much as the sum of their equivalents, in 1 day.
EXERCISES.
45. A merchant borrows i2250 at 8 per cent, for 2
years, with condition to pay before that time as much
of the principal as he pleases. At tho expiration of 9
months he pays jL'80, and 6 months after £70 — leaving
the remainder for the entire terra of 2 years. How
much interest and principal has he to pay, at the end
of that time > ins. J£;i27 16^.
46. I borrow ^£300 at 6 per cent, for 18 months,
with condition to pay as much of the principal before
the time as T please. In 3 months I pay £G0 ; 4 months
after i^lOO j and a months after that £75. How much
principal and interest am I to pay, ut tho end of 18
montiis.? Ans. £,"^9 los.
47. A gives to B at interest on the 1st November,
1804, £6000, at 4^ per cont. B is to repay him with
interest, at the expiration of 2 years —having liberty to
pay before that time as nmch of the principal as he
pleases. Now B pays
£
900
The IGth December, 1804,
The 11th March, 1805,
The P.i'Ui March,
The 17th August,
The 12th February, 1806,
1260v
600
800
1048
How much principal and interest is he to pay on th#
1st Novembor, 1806 ? Ans. £1642 9^. 2if fZ.
48. Ticntat interest £600 the 13th May," 1833, for
24n
INTIRIST.
1 year, at 5 per cent—with condition that the receiver
may diHchargo as much of the principal before the tinm
ns ho pk.as,vs. Now ho pays tlio i)tli July .£^200 ; and
the 1/th beptember i^l50. How nmch principal and
mtorost IS ho to pay at the expiration of the year ?
Ans. £26Q 13s. b^\d. ^
I ^'^' }^}^yv^^ that the pupil, from what he hag
learned of the properties of proportion, will easily un
derstand the modes in which the following rules are
proved to be correct.
Of the principal, amount, time, and rate— given any
three, to find the fourth. ^ ^
Given the amount, rate of interest, and time : to find
the prmcipal — '
^ Rule.— Say as £\Q0^ plus the interest of it, for the
given tune, and at the given rate, is to ^100 ; so ia the
given amount to the principal sought.
^^ExAMPLE.Whiit will produce £862 in 8 years, at 5 per
giv1n'rfe.^'Tf2ir? "'""^ '""^ ^''' ^" ' ^^^ ^' *^«
£140 : £100 : : £862 : J^^^^ =£615 14.. 31^.
When the time and rate are given
n.„^ll!? • ^^^ °^^''"' ^"""^ • ' ^"*®^^'** of £100 : interest of
that oilier sum.
By alteration [Sec. V. 20], this bocomes
th^^mn" '°*"''^* ""^ ^^^^^ '' ■ "^"^ °*^®'' '"'"' • ^°*^''^^*^ ^^
rw v%Ti"° "th« first + the second : the second," &c.
L^ec. V. 2\)] we have —
f^fi^^ i I*' interest : £100 : : any other sum f its in
terest . that sum— which is exactly the rule.
EXERCISES.
49. What principal put to interest for 5 years will
nmount to £402 10.., at 3 per cent, per annum ? Ans,
50. What principal put to interest for 9 years, at 4
percent., will amount to £734 S.. .=^ Ans.
INTEREST.
249
51. Tho amount of a certain principal, bearing inter
est for 7 y(;ars, at 5 per cent., u jL'334 16*. What is
tho j)riucipul ? Ans. £24ii.
1.'5. Given tho time, rato of interest, and principal—
to find tho amount —
lluLK.—Say, as JEIOO is to JCIOO plus its interest for
the given time, and at the given rato, so is tho given
sum to tho amount required.
ExAMPLK.— What will £272 jomo to, in 5 years, at 5 per
C6Ht* f
^'fi}^r (='^1^0f£5x5) is the principal and interest of
*.10U fur 5 years ; then —
£100 : £125 : : £272 : ^~=.£ZiO, the required
amount.
We found by tho last rule that
£iOO+its interest : £100 : : any other sumf its interest :
that sum.
Inversion [See. V. 20] changes lliis into,
£100 : £100fits interest : : any other sum : that other
Bumfits mterest— which is the pi.vBont rule.
EXERCISES.
52. What will £350 amonnt to, in 5 years, at 3 per
cent, per annum ? Ans. £402 10a\
53. What will £540 amount to, in 9 years, at 4 per
cent, per annum ? Ans. £734 8.y.
54. What will £248 amount to, in 7 years, at 5 per
cent, per annum ? Ans. £334 16s.
55. What will £973 4s. 2d. amount to, in 4 years
and 8 months, at 6 per cent. ? Ans. £1245 145. l^d.
56. What will £42 3^. Qid. amount to, in 5 years
and 3 mouths, at 7 per cent. ? Ans. £57 13.?. lOirZ.
16. Given the amount, principal, and rato — to find
the time —
Rule. — Say, as tho interest of tho given sum for 1
year is to th • • • 
juired time.
given interest, so is 1 year to the re
250
INTEREST.
X14 l5. 8rf. (the interest of £281 13s. 4d. for 1 year r21) •
£56 6s Sd.
.^56 Gs. iM. (the given interest)
lequircd number of years.
1:
X14 Is. 8(/.=^' *^«
17.
iience bij
• + . r , .'^^'%> to find the time— Divide the
InZlt '^l^'"^'" ^"'^'^^^^ ^^^ 1 3^^^^' i«to the entire
interest, *ud the quotient will be the time.
in/ln'f •'^'^^''*' *^'" P"°«ipal, and rate beina; civen, the
ntere«t is prcpouonal to the time; the longer the Jme the
more the interest, ^ud the reverse. That is '
ihe interest for one time : the interest for another • •
the former time : the latter. <iuouier . .
^ Hence the interest of the given sum for one year rthe
nterest for o,ic time) : the given interest (the interest of
the same sum for amther time^ • • 1 vpo^ h\Z ,. ^^^^^f.^*
produeedthe former) : thHfi sought tiatXiX^^^^^^^
uuced the latter).which is the rule. ^
EXERCISES.
57. la what time wciild .^300 amount to ^£372, at 6
per cent. > Ans. 4 years, '
58 In what time would £211 5s. amount to ^£230
moniht' ^'' ''''*•• ^^' ^" ^ y^^^ ^^^ 6
59. When would £561 15s. become £719 Os 95^
at 6 per cent. ? Ans. In 4 years and 8 months. * '
60. When would £500 become £599 3s. 4d., at 7 per
""^^V .,r<^'"" ^^ ^ y^^^"« ^n^ 10 months. ^
61. When will £436 9s. 4d. become £571 8s Ud
at 7 per cent. ? Ans. In 4 years and 5 months. ' ' '
the rat^""' *^'^ amount, principal, and time— to find
EuLE— Say, as the principal is to £100, so is the
given interest to the interest of £100— which will give
he interest of £100 at the same rate, and for the same
he rat ' ' ^ *"''''' ^""^ *^^' ^''^*^'^* ^" ^«
INTEREST.
251
Example.— At what rate will £350 amount to £402 10s
in 5 years ?
£350 : £100 : : £52 10s. : ^^^ 10.. x 100
350
=£15, the in
terest of £100 for the same time, and at the same rate
IJien 'j=3, is tbi required number of years.
We have seen [14] that the time and rate being uie same,
£100 : any other sum : : the interest of £100 : interest
of the other sum.
This becomes, by inversion [Sec. V. 29] —
Any sum : £100 : : interest of the former : interest of
100 (for same number of years) .
But the interest of £100 divided by the number of years
wliich produced it, gives the interest of £100 for 1 year—
or, in other words, tlie rate.
EXERCISES.
62. At what rato will c£300 amount in 4 years to
i£372 ? Ans. tj per cent.
63. At what rate will £248 amount in 7 years to
£334 16s. ? Ans. 5 per cent.
64. At what rate will £976 145. 7d. amount in 2 years
and 6 months to £1098 IQs. 4^d. } Ans. 5 per cent.
Deducting the 5th part of*the interest, will give the in
terest of £070 145. Id. for 2 years.
65. At what rate will £780 175. Gd. become £937
Is. in 3 year.1 and 4 months ? Ans. 6 per cent.
66. At what rate will £843 5^. 9d. become £1047 Is.
7<7., in 4 years and 10 months ? Ans. At 5 per cent.
67. ^ At what rate will £43 25. 4JyZ. become £00 75
4Jrf., in 6 years and 8 months } Ans. At 6 per cent.
68. At what rate will £473 become £900 135. fii^Z
in 12 years and 11 months ? Ans. At 7 per cent.
COMPOUND INTEREST.
19. Given the principal, rate, and. time — to find the
amount and interest —
lluLE I. — Find the interest due at the first time of
payment, and add it to the principal. Find the interest
■t
H
r
i 1
1
• t
1 ,
1 1
I ■
m 1
■■
;; i
■
1
1; ,
K i. :. J
. ; ,
ilHiiJ
■i m
m
252
INTEREST.
of that sum, consiJorotl as a new priucipal, and add it
to what it would produce at the next payment. Con
sider that new sum as a principal, itud proceed as
before. Continue this piocess through all the times of
payment.
Example.— What is the compound interest of £97, for 4
years, at 4 per cent, lialfyoarly '?
£ s. d.
97
^ 3 17 7.i is the interest, at the end of 1st half year.
100 17 1\ is the amount, at end of Isfc halfyear.
4 8^ is the interest, at the end of 1st year.
104 18 3'/ is the amount, at the end of 1st year.
4 3 11 is the interest, at the end of 3rd half year.
109
4
2
7
3 is the amount, at the end of 3rd halfyear.
3L is the interest, at the end of 2nd year.
113
4
118
4 14
9 0.} is the amount, at the end of 2nd year.
10 9.^ is the interest, at the end of 5th halfvcar.
4 is the amount, at the end of 5tli luilfyear.
5 is the interest, at the end of 3rd year.
122 14 9 is the amount, at the end of 3rd year.
4 18 2\ is the interest,* at the end of 7th halfyear,
127 12 Hi is the amount, at the end of 7th halfyear,
5 2 1^ is the interest, at the end of 4th year.
132 15 0;i is the amount, at the etui of ith year.
97 is the principal.
And 35 15 0^ is the compound interest of £97, in 4 years.
20. This is a tedious mode of proceeding, particularly
when the times of payment are numerous ; it is, tlune
foro, better to use the following rules, which will be
found to produce the same result —
KuLE II. — Find the interest of £1 for one of the
payments at the given rate. Find the product of so
luany factors (cdch of them cGlfits interest for o!ie
payment) as there arc times of payment ; multiply this
product by the given principal ; and the result v/ill bo
the principal, plus its compound interest for the given
Ur
INTEREST.
253
time. From this subtract the principal, and the remain
der will be its compound interest.
Example 1. — What is the compound interest of £237 for
3 years, at 6 per cent. '?
£0Q is the interest of £1 for 1 year, at the given rate ;
and there are 3 payments. Therefore £106 (^£l\£0\j) ig
to be taken 3 times to form a product. Hence lOGxlOOx
l06x£237 is the amount at the end of three years; and
l0Gxl06xl06x£237— £237 is the compound interest.
The following is the process in full —
£
106 the amount of £1, in one year.
1'06 the multiplier.
11236 the j,mount of £1, in two years
lOG the multiplier.
1191016 the amount of £1, in three years
Multiplying by 237, the principal,
£ s. d.
wo find that 282270792=282 5 5 is the amount •
and subtracting 237 0, the principal,
we obtain 45 5 5 as the compound interest>.
Example 2. — What are the amount and compound inte
rest of £79 for 6 years, at 5 per cent. 1
The amount of £1 for 1 year, at this rate would be £10.5.
^ Therefore £105 X 105 X 105 X 1 '05 X 105 x 105x79 is the
amount. &c. And the process in full will be —
£
105
105
11025 the amount of £1, in two years.
11025
11^1551 the amount of £1, in four years.
11025
134010 the amount of £1, in six years.
£ s. a.
£10586790=105 17 4 is the required amount.
79
And 26 17 4 is the rcqnired interest
M 2
2.11
tNTEKKST
Example o. — Whataro the aiTJount, and compound interest
of £27, for 4 years, at £2 10s. per cent. Iialfyearly.
The anionnt of XI for ono pnyinont is X102:>. Therpfore
ClOiry X I 025 X 1 025 X 102o x 1025 x 1025 x 1025 x
I 025 X27 is the amount, &o. And the process in full will be
£
1025
1025
1 05003 the amount of XL in one year,
105003 "^
T 10382 the amount of XI, in two years.
110382 ■
121842 the amount of XI, in four years.
27 , ^
£ s. (17
X328U734=32 17 11 is the required amount.
: 27
And 5 17 li is the required interest.
21. Rule IIT.— Find by the interest table (at the end
of the treatise) the amount oi £,\ at the .f^iven rate, and
for the given number of payments ; multiply this by tho
given principal, and the product will be the required
amount. Prom this product subtract the principal, and
tho remaiader will be the required compound interest.
Example.— AVhat is the amount and compound interest
of X47 lOi. for 6 years, at 3 per cent., halfyearly '? <
X47 10,y.=X475.
We find by the table that
X14257G is the amount of XI, for the given time and rate.
475 is the ■ .altiplicr.
~rr~~ ^ •"'• ''•
677230=67 14 5'' is the required amount.
47 10
And 20 4 5^ is tho required interest.
22. Tlule r. requires no explanation.
li,KAso\' OK F.uLK II. — Whon the time and rate are Mie
Bfvme, ivfo priicipnls are proportional to their corresponding
amounts. Thcrofore
£1 (one principal) : £1 03 (its corresponding amount) :
£10G (.another principal) : £100 X lOG (its corresponding
amount).
IXTF.UKST.
2i?5
Ileneo the amount of £1 for two ycirs, is £106xl'06—
or the product of two factors, each of them the amount of £1
for one yeur.
Again, for similar reasons,
£1 : £106 :: £l06Xl0G : £1 OGXlOGXlOG
Hence the amount of £1 for three years, la £l'06xl'06Xl*06—
or the product of three factors, each of them the amount of
£1 for one year.
The same reasoning would answer for any number of pa^'
ments.
The amount of any principal will be as much greater than
the amount of £1, at the same rate, and for the same time, as
the principal itself is greater than £1. Hence we multiply
the amount of £1, by the given principal.
Rule III. requires no explanation.
23, When the decimals bejonie numerous, we may
proceed as already directed [Sec. II. 58].
We may also shorten the process, in many cases, if
we remcTaber that the product of two of the factors
multiplied by itself, is equal to the product of four of
them ; that the product of four multiplied by the pro
duct of two is equal to the product of six ; and that the
product of four multiplied by tho product of four, is
equal to the product of eight, &c. Q'hus, in example 2,
M025 (=l05xl05) xi1025=105xl05xl05xl 05.
EXERCISES.
1 . What arc the amount and compound interest of
£91 for 7 years, at 5 per cent, per annura ^ Ans. £12S
05. l]d. is the amount; and .£37 05. lit/., the com
pound interest.
2. What are the amount and compound interest of
£142 for 8 years, at 3 per cent, halfyearly.? Ami.
£227 175. 4ld, is the amount ; and £85 175. 4^^., the
compound interest.
3. What are the amount and compound interest of
£63 55. fi , ars, at 4 per cent, per annum ? Ans.
£90 05. 5f//.'is the amount; and £26 155. Sfc/., tho
compound ixiterest.
4. What are the amount and compound interest of
£44 05. dd. for 1 1 years, at 6 per cent, per annum .'*
f\'
256
INTEREST.
Ans. £84 Is. tul. is the amount; aud £39 155. Sd
the compound interest. "'
i). What are the amount and compound interest of
£^2 4s. ^d. for 3 years, at £2 \0s. per cent, halt
l ^l^l t''' ^^^ ''• ^■^'^' '^ *^« ^^riouxii and JC5
^s. lUif/., the compound interest.
r.o^"i ^^'^^^^ ^^^ *^^" amount and compound interest of
^971 0,; 2\d. for 13 y.^ars, at 4 per cent, per annum }
Ans in 616 15.S. 115^^. is the amount; and £645 Ids
Jid.j the compound interest.
24. Given the amount, time, aud rate— to find the
pnncipal ; that is, to find the present icorth of any sum
to be cire hereafter— a certain rate of interest being
allowed for the money now paid.
lluLE.— Find the product of as many factors as there
are tunes of payment— each _ of the factors bein^ the
amount of £\ for a single payment ; and divide this
proau<"t mto the given amount.
Example.— What sum would produce £834 in 5 years,
at per cent, compound iateresf?
TJio amount of XI for 1 year at the given rate is £105 ;
}oXl'^^^,^'^''^X^"0^' ^^^"«^ (according to the table) is
i:831M27G28=i:G53 9s. 2,^./., the required principal.
25. Reason of thk Rui.k.—Wc have seen [21] that the
nmoxmt of nny sum is equal to the amount of &\ (for the sanu
tm.e, and at the same rate) multiplied by tlie principal ; that is,
the !;L;uroV,£ L ''' ^'^^■^^ pHnoipal==thc given ^incipalX
r,nl^JI.%J'™v ^'w,'^' ^''''? '^"''^ quani.tics by the same
number [bee. V. b], the quotients will be equal. Tl.croforc—
_ Ihe amount of :'.o given principal f the amount of £l=thc
given prmcipalxthe amount of £lMhe amount of &\. Tliat
r' • 1 ] T^T^ ""^ ^^"^ ■?'^'"'' princiijal (the given amount)
divided by the amount of £1, is equal to the principal, or
quantity required— which is the rule. i ^^ »^
EXERCISES.
7. What ready money ought to be paid for a debt of
X629 176. \\\d., to be duo 3 years hence, alluwin'
S i)er cent, compound interest } Ans. j£^500. °
INTRRERT.
257
8<f.,
H. mat principal, put to interest for G years, would
0. What sum u'ould produce ^£742 Ids UUl in 14
years, at 6 per cent, per annum ? Ans. ^32s' 12.. 7d.
10. ^^ liat IS .€495 19.. llr/., to be due in IS years,
il71 IT'sfd ^^'^^"^'*^^"^'^^' ^^'^^th at present. ^^«.:
tlio^timo^*'" ^'"^ P''"'cip<'^^ rate, and amount— to find
^ Rule I.—Divide the amount by the principal: and
mto the quotient divide the amount of £1 for one my^
ment (at the given rate) as often as pcssiblethe number
ol times the amount of £1 U, been used as a divisor,
will be the required number of payments.
0\i''il'T'~^'' jvhat time Avill £92 amount to £100 13..
V^d., at 3 per cent, halfyearly '?
XlOG 13.. O'v/.^.:e92=ll.o927. The amount of £1 for
one payn.ont i. ,£103. Vnit 115927  103  11255 •
1 • 12.),) ^ 103 == ln'J272 • I0'i'^7'> • 10'^ — i n^^o A
10009^103^1.03; lS3il^j:;i\(;eIale^.i t^
as a divisor tiinco; therefore the time is 5 pavmenta. or
S^^vTo^r'^""'^ *''"'" ^''^^ ^'" ^ ron)ainder after dWid
Dip, \>y i Uo, tec. as often as possible
In explaining the method of tinding the powers and roots
moni7n?n'^"^''f •^■' ''i "^'^'^ hereafter, notice a shorter
method of aseertan.ing ],cnv ofton the amount of one pound
can Ije used as a divisor. '■
27. Rule H.—Divide the given principal by the
given amount, and ascertain by the interest table in how
iiumy pnyments £1 would be equal to a quantity nearest
to the quotient— considered as pounds : this will be tho
rcquured time.
KxAMPi.K.—In what time will £50 become £100, at G
per cent, per atmum compound interest ?
£100150=2.
^i^«o^^o''M''^ 1^'? ^f^'^'' *'^''* ^" 1^ years £1 will become
ro moo '1'1' ■' ^^'' ' '^'"^ ^" ^ y^^'' ''"It it ^^'ill l^ecome
ii '? .; ^; ""^\.'''^ ^^y^^^^ than 2. Tho answer nearest to
the truth, thereluro. is 12 years.
li^Si
^8
INTKUr.ST,
28. Rkason ok lliTLK I. — TIio given amount is [20] cquul
to the givon principal, nuiltiplioil by a proiluct wlucii containa
as many factors as there arc tijiies of payment— each factor
being the anioiint of i^l.fur one payment. Hence it is evi
dent, that if we divide the given amount by tlie given prin
cipal, we must have the product of these factors ; and tiiat, if
we divide this product, and tlio succesHivo quotients by one
of the factors, wc shall ascertain their number.
l^EAsoN OF llui.E 11. — We can find the required number
of factors (eacli tlie amount uf £1), by ascertaining how often
the amount of .£1 may bo considered us a factor, withuut
forming a product tmich greater or less than the quotient
obtained when we divide the given amount by the given
principal. Instead, however, of calculating for ourselves, we
may have recourse to tables constructed by those who have
already made the necessary multiplications — which saves much
trouble.
29. When the quotient [27 J is greater than any
amount of £\^ at the given rate, in tlie table, divide it
by the greatest found in the table ; and, if necessary,
divide the resulting quotient in the same way. Continue
the process until the quotient obtained is not greater
than the largest amount in tlie table. Ascertain wliat
fimiber of jxnjments corresponds to the last quotient,
and add to it so many times the largest nniuler of pay
vients in the table, as the largest amount in the table
has been used for a divisor
ExAMPLK. — When would £22 become X535 12s. O^d.,
at 3 per cent, per annum '?
£535 12s. OJri.^ 22=2434500, which is greater than any
amount of £1, at the ^iven rate, contained in the taljle.
2434560f4383l) (the greatest amount of £1, at 3 per cent.,
found in the ta1)lc)=5'55339 ; but this latter, also, is greater
tluxn any amount of £1 at the giv<n rate in tlie tables.
555339i4'383'J=l'2(iG77, which is found to bo the amount
of £1, at 3 per cent, per payment, in 8 payments. We
have divided by the highest amount for £1 in the tables, or
that corresponding to iifty payments, twice. Therefore, the
required time, is 50j50f8 payments, or 108 years.
EXERCISES.
11. When would £14 6^. 8^. amount to i218 2s. 8^d.
at 4 per cent, per annum, compound interest ? Ans.
In 6 years.
INTEUKST.
259
12. Wlion would jer)4 25. 8^/. amount to £76 35. 5d.^
Hi 5 per cent, per annum, compound interest .'* Ans.
Tn 7 years.
13. In wliat tinu! would £793 ().?. 2]f/. become J21034
135. IOJyZ., at 3 per cent, halfyearly, compound interest ?
Ans. hi 4^ years.
14. '\Vhcn would £100 become £1639 7.?. 9J., at 6
per cjLt. haltyearly, compound interest .'' Ans. In 24
years.
QUESTIONS.
1. What is interest .? [IJ.
2. AVliat is the diffLjrenco between simple and com
pound interest ^ [1].
3. AVhat are the principal, rate, and amount ? [1].
■ 4, How is the simple interest of any sum, for 1 year,
found.? [2 &c.].
5. How is the simjilo interest of any sura, for several
years, found } [5].
6. How is the interest found, when the rate consists
of more than one denomination ? [4].
7. How is the simple interest of any sum, for years,
months, &c., found ? [6].
8. How is the simple interest of any sum, for any
time, at 5 or 6, &c. per cent, found .? [7].
9. How is tlie simple interest found, when the rate,
number of years, or both arc expressed by a mixed
number ? [9J.
10. How is the simple interest for days, at 5 per cent.,
found .? [10].
1 1 . How is the simple interest for days, at any other
rat«, found ? [H].
12. How is the simple interest of any sum, for months
at f) per cent., found t [12].
13. How is the interest of money, left after one or
more payments, found ? [13].
14. How is the principal found, when the amount,
rate, and time are given .'' [14].
1"), How is the" amount found, when the time, ratn,
and principal arc given ? [15].
■hM
2G0
DISCOUNT.
10. ITaw k tliG timo r.uiad, when the amount, prin
cipal, II ud rate are given ? [10 J.
17. How is the rate found, when tlio amount, priuci
pal, and timo arc given ? [18].
18. How are the amount, and compound interest found^
wlion the principal, rate, and time are given ? [iDj.
I'J. llow is the present worth of any sum, at com
poiuid interest for any time, at any rate, found > [24.
20. How is the time found, wlirm t)ie principal, rate
of compound interest, and amount are given .? [26j.
DISCOUNT.
30. Discount is money allowed for a sum paid before
it is due, and should be such as would be produced hy
what iii paid, were it put to interest from the time the
payment is, until the time it ou,o/ii to he made.
The presoit loorth of any sum, is that which
would, at the rate allowed as discount, produce it if
put to interest until the sum becomes due.
'M. A bill is not payable until thiee days nfter the
time mentioned in it ; those are called days of grace.
TIuis, if the time expires on the 11th of the month, the
bill will no^. be payable until the 14th— except the latter
falls on a Sunday, in which case it boconies payable on
the preceding Saturday. A bill at 91 days will not be
duo until the 04th day after date.
32. WHicn goods are purchased, ascertain discount is
oft^Mi allowed for prompt (immediate) payment.
The discount generally take;, is larger than is sup
posed. Thus, lot what is allowed for paying money
one year before it is duo be 5 per cent. ; in°ordinary
circumstances ^£95 would bo tho payment for .£100.
But £\)b would not in one year, at .5 per cent., produce
more than i299 15.v., which is less than £100 ; the erior,
however, is inconsiderable when the time or sum is small
Hence to find the discount and present worth at any
rate, we may ge.nerallu use the following —
DItCOUNT.
901
2?.. Hulk. — Find the interest for the sura to be paid,
at tlu; dificouut uUowod; consider this ns discount, and
(hduct it from wha( is due ; the romaindt • will be tlus
required present woiLh.
Example. — £<V"' will ^ ; duo in 3 months ; what should b*
allowed on inui, lo payment, the discount being at the
rato of G per cent, per annum 1
The intorest on £(i2 for 1 year at G per cent, per annum
is cC3 1 ' 4'l(l. ; and for 3 months it is IHs. l^d. Therefore
jEi02 miuua iSs. 7ti.=J(iGl Is. 4^(i., u the required present
worth.
.34. To find the present wortli acrMrakly —
]Ii;le. — Say, as .£100 plus its interest for the given
tiuio, is to iilOO, ^ ) is the given pum to the required
present worth.
TCxAMPLE. — What wnuld, at present, pay a debt nf XI 12
to he due in (J months, b per cent, per annum disooont being
allowed ?
jC
£ £ s. £ £ i()f) V 142 ^ •'• ^•
1025 (100 f2 10) : 100 : : 142 : — xi^4~=^^^ ^^ ^
This is merely a question in a rule already given [14].
.1 I
i
EXERCISES.
1. What is the present worth of ^2850 15i., payable in
one year, at G per cent, discount > Ans. £802 lis. lO^d
2. What is the present worth of £240 10.?., payable
in one year, at 4 per cent, discount ? Ans. £231 5^.
3. What is the present worth of £550 10s., payable
in 5 years and 9 months, at 6 per cent, per an. discount >
Ans. £409 55. loyi.
4. A debt of £1090 will be due in 1 year and 5
months, what is its present worth, allowing 6 per cent,
per an. discount ? Ans. £1004 12s. 2d.
5. What sum will discharge a debt of £250 175. 6c?.,
to bo due in 8 months, allowing 6 per cent, per an.
discount .=" A71S. £241 45. 6 J J.
6. Wiiat sum will discharge a debt of £840, to be
duo in 6 montlis, allowing 6 per cent, per an. discount ?
A71S. £815 IO5. HUl
IMAGE EVALUATION
TEST TARGET (MT2)
1.0
.1
9^ IM 111112 2
If 1^4 lit
 1^ lilM
il.25
1.4
1.8
1.6
<^
/a
'e.
^/^
'V/
■'^A
<Pl
^7 ^"4
#
!m.
Phntnorpnlrlr
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 8724503
#
:1>^
\
:\
"Q
\
«
<f
^
#
^'""^«?:^
^^^'
.: W ■
o
V <^ ^ #/
4f^
«.
i^
^
^
:\
\
6^
262
15ISC0UNT.
7 What ready money now will pay a debt of £200,
to bo due 127 days hence, discounting at 6 per oent
per an.? Am. £ldo iSs. 2}d. f « ut.
8.^ \Vhat ready money now will pay for ^1000, to be
9 A bill of £150 105. will become due in 70 days
what ready money will now pay it, allowing 5 per cent
per an. discount ? Ans. dei49 Is. bd
10. A bill of £140 10,. will be due in 76 days, what
ready money will now pay it, allowing 5 per cent, per
an. discount > Ans. £139 1,. O^'Z. ^
11. A bill of £300 will be due in 91 days, what wiU
^oTu?^^ ^\ al owing 5 per cent, per an. discount .? Ans.
jb29D <ys. l^d.
12. A bill or £39 5^. will become due on the first
ot beptember, what ready money wUl pay it on the
^^£38 iS 15"^^' ^^^''''''° ^ ^'' "'''*• P"" ^" •
13. A bill of £218 3.9. SicZ. is drawn of the 14th
August at 4 months, and discounted on the 3rd of Oct •
what IS then its worth, allowing 4 per cent, per an!
discount .? Ans. £216 Ss. \id.
14 A bill of £486 185. 8^. is drawn of the 25th
March at 10 months, and discounted on the 19th June
what then is its worth, allowing 5 per cent, per an'
discount.? Ans. £412 9s. U^d. e ■
15. What is the present worth of £700, to be due in
9 months, discount being 5 per cent, per an. > Ans
£674 135. 11^^. ^ p u. . Jins.
16. What is the present worth of £315 12, 41^/
payable in 4 years, at 6 per cent, per an. discount"?
Ans. £254 lO.e. 7]d.
17 What is the present worth and discount of £550
105. for 9 months, at 5 per cent, per an. } Ans. £530
125. Q\d. is the present worth; and £19 175. lli^;
s the discount. * *
18. Bought goods to the value of £35 135. 8^. to be
Daul m 294 days; what ready money are they now
ivorth, 6 per cent, per an. discount being allowed ?
Ans. £31 05. 9^^/. ^
COMMISSION.
263
19. If a legacy of £600 is left to me on the 3rd of
May, to be paid on Christmas day following, what must
I receive as present payment, allowing 5 per cent, per
an. discount.^ Ans. i:i581 4s. 2}d.
20. What is the discount of £756, the one half pay
able in 6, and the remainder in 1.2 months, 7 per cent,
per an. being allowed ? Ans. £37 I4s. 2\d.
21. A merchant owes £110, payable in 20 months,
and £224, payable in 24 months ; the first he pays in 5
mouths, and the second in one month after that. What
did 1)0 pay, allowing S per cent, per an. } Ans. £300.
QUESTIONS FOR THE PUPIL.
1. What is discount .? [30].
2. Wliat is the present icorth of any sum } [30].
3. \s'\\iii QXQ. days of grace] [.?1].
4. How is discount ordinarily calculated } [33]
5. How is it accMrately calculated > [34] .
COMMISSION, &c
3.5. Commission is an allowance per cent, made to a
person called an agent., who is employed to sell goods.
Insurance is so mucli per cent, paid to a person who
undertakes that if certain goods arc injured or destroyed,
he will give a stated sum of money to the owner.
Brokerage is a small allowance, made to a kind of
agent called a broker, for assisting in the disposal of
goods, negotiating bills, &c.
36. To compute commission, &c. —
Rule. — Say, as £100 is to the rate of commission, so
Is the given sum to the corresponding commission.
ffc Example. — What will be the commission on goods worth
£437 56. 2</., at 4 per cent. ]
£100 : £4 : : £437 5s. 2d. : l^l^iJIii?^ = £17 9^.
100
9i'<Z., the required couimissiion.
V7.V 37. To find what insurance must be paid so that, if
the goods are lost, both their value and the insurao'io
paid mr.y be recovered —
264
COMMISSIOI*
Rule.— Say, as £100 minus the rate per cent, is to
eClOO, so is the value of the goods insured, to the
required insurance.
Example.— What sum must I insure that if goods worth
i.4UU are lost, I may receive both their value and the in
surance paid, the hitter being at the rate of 5 per cent '*
£95 : £100 :: £400 : ^122^0=^421 1. 0^^
If £100 were insured, only £95 would be actually received,
since £5 was paid for the £100. In the example, £421 Is Ohd
are received; but deducting £21 Is. OU, the insurance, £400
remains,
EXERCISES.
1. What premium must be paid for insuring goods
to the amount of £900 15s., at 2^ per cent, f A7is
£2,2 105. 4ir/. '  f
2. What premium must be paid for insuring goods
to the amount of £7000, at 5 per cent. ? Ans. £350
3. What is the brokerage on £976 175. 6d., at 55.
per cent. > Ans. £2 Ss.'lQid.
4. What is the premium of insurance on goods worth
£2000,^ at H per cent. ? Ans. £150.
5. ^Vlmt is the commission on £767 145. 7d , at 2i
pcrcent. .? A7is. £19 3s. lO^d. ^
y,^',^^'''^ ^'^'^^ ^^ *^^e commission on goods worth
i'J71 145. 7rf., at 5.5. per cent. ? Ans. £2 8s. 7^d
7. What is the brokerage on £3000, at 25. 6^. per
cent. ? Ans. £3 15s. ^
S How much is to be insured at 5 per cent, on goods
worth £900, so that, in case of loss, not only the value
ot the goods, but the premium of insurance also, may bo
repaid ? ^ Ans. £947 75. 4/^. ' ^
9 Shipped off for Trinidad goods worth £2000, how
much must be insured on them at 10 per cent., that in
case of loss the premium of insurance, as well as their
value, may be recovered ? Ans. £2222 45. dhd.
QUESTIONS FOR THE PUPIL.
1
What is commission ? [35],
2. What is insurance ? [35].
3. What ia brokerage ? [35]
PURCHASE OF STOCK.
265
0?rf.
at24
4. IIow are commission, insurance, &c., calculated?
[36].
5. How is msnrancG calculated, so that both the in
surance and value of the goods may be received, if tho
latter are lost ? [37] .
PURCHASE OF STOCK.
.>o.
Stock is money borrowed by Government from
individuals, or contributed by merchants, &c.,^ for the
purpose of trade, and bearing interest at a fixed, or
variable rate. It is transferable either entirely, or in
part, according to the pleasure of the owner.
If the price per cent, is more th;in £100, tlie stock in
question is said to be ahave^ if less than i^lOO, helow " par."
Sometimes the shraes of trading companies are only
gradually paid up ; and in many cases the whole price
of the sliare is not demanded at all — they may be ^£50,
£100, &c., shares, while only £5, £10, &c., u.ay have
been paid on each. One person may have many shares
When the intesest per cent, on i\\Q money paid is con
.sidera1)lo, stock often sells for more than what it origi
nally cost; on the other hanu, when money becomes
more valuable, or the trade for which the stock was
contributed is not prosperous, it sells for less.
39. To find the value of any amount of stock, at any
rate per cent. —
EuLE. — Multiply the amount by the value per cent.,
and divide the product by 100.
ExAMPi.K.— When £G'J \ will purchase £100 of stock, what
will purchase £G42 ?
£G42x69j
100
=£443 15s. lid.
It is evident that £100 of stock is to any other amount of
it, as the price of tho former is to that of tho hxttor. Tims
£100 : £612 :: £69 .\ : ^il^^.^
100
EXERCISES.
1. What must be given for £750 16i\ in the 3 per
cent, annuities, when £64 j will purchase £100 .? Ans.
£481 95. O^^V^
■I
iii*(
2Rf)
EQUATION OP PAYMENTS.
2. What must be given for ^£1756 Is. 6d. India stock,
when £]U6l will purchase dt^lOO ? Ans. £3446 17s. i<^d
3. What is the purchase of ^29757 bank stock,' ai
J212oA per cent. ? Ans. jei2257 4^. 7^d.
QUESTIONS.
1. What is stock .? [38].
2. When is it above, and when below " par" ? [38],
3. How is the value of any amount of stock, at an^
rate per cent., found ? [39].
EQUATION OF PAYMENTS.
40. This is a process by which we discover a time,
when several debts to be due at dij'ereni periods maybe
paid, (il once, without loss either to debtor or creditor
lluu.:.Multiply each payment by the time which
should elapse before it would becoie due ; then add
the products together, and divide their sum by the' sum
01 the debts.
Example 1. A person owes another £20, payable in 6
months; i 50 payable in 8 months; and X90 payable in
U months. At what time may all be paid together, without
OSS or gam to either party '? * o )
il jl
20 X 0= 120
/30x 8= 400
_90x 12=1 080
IGU 1GO)TOOO(10 the required number of mor'^g.
160
. ExAMPLK 2.— A debt of £450 is to be paid thus : £100
immediately, £300 in four, and the rest in six months \V lien
Bhould it be paid altogether ?
£
100
300
_50
450
£
X 0=
X 4=1200
X 6=^03^
450)1500(31 months
1350
"iso
450
EQUATION OP PAYMKNTS.
•267
41. Wo liavG (according to a i^rinciplo fonnorly used
[13]) reduced each debt to a sum which would bring the
same interest, in one month. For G times i^20, to be due
in 1 month, should evidently produce the fsame as £20, to
be due in G months — and so of the other debts. And the
interest of j£lGOO for the smaller time, will just be equal to
the interest of the smaller sum for the larger time.
EXERCISES.
1. A owes B jeeOO, of which £200 is payable in 3
months, £150 in 4 months, and the rest in 6 months ;
but it is agreed that the whole sum shall be paid at
once. When should the payment be made ? Ans. In
41 months.
2. A debt is to be discharged in the following man
ner : I at prcijent, and ^ every three months after antil
all is paid. What is the equated time ? Ans. A\
months.
3. A debt of £120 will be due as follows : £50 in
2 months, £40 in 5, and the rest in 7 months. "When
may the whole be paid together 't Ans. In 4^ months.
4. A owes B £110, of which £50 is to be paid at
the end of 2 years, £40 at the end of 3^, and £20 at
the end of 41^ years. When should B receive all at
once .'' Ans. In 3 years.
5. A debt is to be discharged by paying ^ in 3 months,
i in 5 months, and the rest in 6 mouths. What is the
equated time for the whole .'' Ans. 4 months.
QUESTIONS.
1. "What is meant by the equation of payments }
2. What is the rule for discovering when money, to
be due at different times, may be paid at once } [40] .
! ■
' ii
i.
208
SECTION VIII.
EXCHANGE, &c.
1. Exchange enables us to find what amoimt of the
inoncy of one country is equal to a given amount of the
money of another.
Money is of two kinds, real— or coin, and imaginary—
or money of exchange, for which there is no coin ; as,
lor example " one pou7id sterling."
The par of exchange is that amount of the money
of one country aduallt/ equal to a given sum of tho
money of another ; taking into account the value of
the metals they contain. Tho mirse of cxchan<rc if^
that sum which, in point of fact, would be allowed
for it.
2. When the course of exchange with any plac6 is
a^ove ;' par," the balance of trade is against that place.
Thus if Hamburgh receives merchandise from London
to the amount of ^£100,000, and ships off, in return, goods
to the amount of but c£50,000, it can pay only half what
It owes by bills of exchange, and for the remainder must
obtain bills of exchange from some place else, giving
for them a premium— which is so much lo?.t. IJut the
exchange cannot be much above par, since, if the pre
mium to bo paid for bills of exchange is high, tho
merchant will export goods .it loss profit ; or Tie will
pay the expense of transmitting aixl iusuriu*' coin, or
bullion. °
3. The nominal value of commodities in these countries
was from four^ to fourteen times less formerly than at
present ; that is, the same aiJ^ount of money would then
buy much more than now. We may estimate the value
of money, at any particular period, from the amount of
corn It would purchase at that time. The value of
money fluctuates from the uature of the crops, the statu
of trade, &o.
KXCHANGE.
209
111 ; a.s,
la cxcliango, a variublo is given for a fixed sum ; ihiin
LonJf)!! receives difioreut values for £1 from diliereut
countries.
Agio is the dilTerence wbieli there is in some places
between the cwrreiU or msk money, and the uxkange
or hank money — which is finer.
The following tables of foreign coins arc to be mad'.'
familiar to the pupil.
FOREIGN MONEY.
MONEY OF AMSTERDAM.
Flemish Money.
• . make 1 groto or penny.
• . • 1 stiver.
• 1 florin or guilder
Penningii
16 or
320
800
1920
giote«
40 or
100
240
stivers
20
50 or
120 or
guilders
2i
6
1 rixdollar.
1 pound.
rfenningg
6
72 or
M40
g rotes
12
MONEY OF HAMBURGH.
Flemish Money.
I • . make 1 grote or penny
lings
1 skilliug.
1 pound.
Ffenmnga Tenco
12 or 2
192
884
676
I skilli
240 orf 20
Ilamhv,rgh Money.
make 1 scliilling, equal to 1 stiver
1 mark.
32 or
64
96
schillings
16
32 or
48 or
marks
2
3
1 dollar of exchange.
1 rixdollar.
We find that 6 scliillings=l skilling
Hamburgh money is distinguished by the word " Harabro."
" Lub," from Lubec, where it was coined, was formerly used
for tliis purpose ; thus, '• one mark Lub."
Wo exchange with Holland and Flanders by the pound
Bt^cling.
N
JiU
rM
i>70
KXCIIAWaE.
KRKJSCn MONKY.
Dernioi!*
12
Accouula wcro Ibnuerly kept in livrus, &c.
210 or
720
Centimes
10
sous
20
make 1 sou.
1 livre.
livres
60 or I 8
1 ecu or crown
AcoountH are now kept in francs and centimes.
• . make 1 dccime.
dccimei
100 or I 10 .
81 livrea=80 franca.
1 franc.
llocs
400
1000 or
4800
PORTUGUESE MONEY.
Accounts are kept in milrees and recs.
«... make 1 crusado.
crnsadoi
2i .
12 ... .
1 milree.
1 moidore.
SPANISH MONEY.
Spanish money is'of two kinds, plate and vellon ; the latter
being to the former as 32 is to 17. Plate ia used in exchange
with us. Accounts are kept in piastres, and maravedi.
Maravedies
84
make 1 real.
272 or
1088
375
reals
8
piastres
32 or I 4 .
1 piastre or piece of eight
1 pistole of exchange.
. 1 ducat.
AMERICAN MONEY.
In some parts of the United States accounts are kept in
dollars, dimes, and cents.
Cents
10 . . , , , make 1 dima
idimea
10 . . . . . 1 dollar.
In other parts accountg are kept in pounds, shillings, and
pence. Those are called currency, but they ar« of much less
yalue than with ua, paper money being usad.
Pf<«nninjj!i
12
KXCHANtiS
DANISH MONET.
i?71
make 1 skill! u;;.
102 or
nkillin
10
K»
marki
1152 OC or (} .
amburgU marks.
VKNETIAN MONEY.
Dnnari (the plurnl ordqnaro)
12 .... innke 1 soldo,
soldi
210 or JiO .
liro golili
1188 121 or "6 4
l'J20 100 8 . .
1 mark.
1 rixdollar
1 lira.
1 ducftt current.
1 du(^t ellcctivo
AUSTRIAN MONEY.
rfcmiiiigs
4
210 or
800
Grains
10
oroutzers
00 __.
fioring
90 or I U
NEAPOLITAN MONEY.
cailius
100 or I 10
mako 1 croutzsr
1 florin.
1 rixdollar.
make 1 carlin.
1 ducat rtt,A9
MONEY OF GENOA.
Lire soldi
4 nnd 12 make 1 scudo di cambio, or crown of exchange.
10 nnd 14 1 scudo d'oro, or gol I crown.
Dcnari di pe/.za
12
OF GENOA AND LEGHORN.
mako 1 soldo di pozza.
I soldi di pez/a
20
f)cnarj di lira
12
240 or
1380
soldi di lira
20 .
110 or 1 5
SWEDISH MONEY.
1 pezza of 8 reals,
make 1 soldo di lira.
Fonnings, or oers
12
Iskillingi
48
1 lira.
1 pezza of 8 reals
make 1 skilling.
1 rixdoUai
I
f ■
ft!
ill
ii i
I
t73
CXOIANOE.
RUSSIAN MONEY.
160
mnko 1 ruble.
EAST
niuko 1 rupco.
DIAN W0NE1
Towriei
Kiinoci
100,000 . . . iiao.
10,000,000 . . 1 croro.
The cowrie is a small ehoU found at the MaMivuH, and near
Anj;ola : iu Africa about 5000 of them pass for a pound.
The rupcq lias different values : at (Juloutta it is 1;». 11 j,/.
tbo Sicca rupee is 2s. OU. ; and the current rupee 2.*.— if wo
divide any number of tlicse by 10, we change them to pounds
of our money; the Boinbny rupee is 2s. {5^/., &c. A sum of
Indian money is expressed as follows; 588220, which means
5 la(!3 aiid«8220 rupees.
d. To rcduco bank to current money —
Ki'i.K.— Say, as J2100 is to JL'IOO + the agio, so is
the given amount of bank to tlio recjuired amount of
current money.
KxAMi'LK. — How many c;uildor.«<, current munoy, arc equal
to 403 ouildors, 3 stivers, and 13!jt ponnings banco, a^io
being 4^' < .0 a
1<J'^ : lO^ : : 403 g. 3 st. 13«4 p. : 1 '
7 7 20 ^"^
TOO
05
733
1)203 stivers.
10
45500
14S221 pcmiinga.
Multiplying by 05, and adding 04 to tho
will give 0034429 I'l'^^li'^t,
]Multi])lying by 733
and dividing by 45500)T0020lio4r)7
will give 155200 penniugs.
10)155209
20 )9700 9
And 485 g. O"^ d^tH p. is the amount sought.
5. We multiply the first and second terms by 7, and add tha
numerator of the fraction to one of tlie products. This is tlie
same thing as reducing these terms to fractions liaving 7 for
their denominator, and then multiplying them by 7 [Sec. V. 29]
For the same reason, and in tho same way, we multiply the
first and third terms b/ 65, to banish tho fraction, without
aeitroying the proportion.
eXCHANOE.
273
TIic remainder of tho process i« nccording to Iho rulo of
lnoporUofi [Soo. V. 1)1]. VVc roduco tho nnawcr to pcnaingH,
BtiverH, mill jjniMoia. •
EXKHCIHF.S.
1. R<m1iico n7l ^•uildofH, 12 Htivors, banlf monoy, to
cm rent money, agio being 4i per cent. ? Am. 31)2 g.,
5 St., :},V, p.
2. llecluce 4378 guililers, «< stivers, bank money, to
current money, agio being 4* per cent. ? Avs. 4577 «.,
17 St., r^Vs p.
3. Ueduee S73 guilders. 1 1 stivers, bank money, to
current money, agio being 4 J per cent. ? A7is. UIG g.,
2st., HJap.
4. llediicc 1012 guilders, bnnk monoy, to current
money, agio being 4i per cent. ? Am. 1722 g., 14Ht.,
lOA p.
6. To reduee current to Itank money —
_ liiu.i:. — Say, as JUlOOfthc agio is to JCIOO, so Is tlio
given amount of current to the required amount of
baidc money.
ExAMi'i.K. — How much bank money is thcro in 485 guil
ders and ^J'ii'iol pennings. agio being 4^' i
104?
7
733
4550c
100
7
700
g. St. p.
20
33351500
yioo
10
15520!)
Multiplying by 45500 tho denominator,
7002009500
and adding 25957 tho numerator,
we get 7002035457
700
33351500)4943424819900
Qu(7tIenl~lT822 1 fl
10) 1 48221;; I
20)9203
403 3 13^^ is tho amount soug*i!
i
I
274
EXCHANttE.
EXERCISES.
5 Reduce 58734 gi^lders, 9 stivers, 11 penningR,
current money, to Lank money, agio being 4^ per cent. ?
Ans. 560P6 g., 10 St., llJfi p.
6. lieduce 4326 guilders, 15 pcnnings, current money,
to bank money, agio being 4f per cent. ? Ans. 4125 g.,
13st.,2ip.
7. Eeduce 1186 guilders, 4 stivers, 8 pennings, cur
rent, to bank money, agio being 4f per cent. } Ans
1136 g., 10st.,0iff p.
8. Keduce 8560 guilders, 8 stivers, 10 pcnnings,
current, to bank money, agio being 4i per cent. .
Ans. 8183 g., 19 st., 5fi3. p.
7. To reduce foreign money to Ikitish, &c. —
BuLE. — Put the amount of British money considered
in the rate of exchange as third term of the proportion,
i^^' value in foreign money as first, and the foreign
money to be reduced as second term.
Example 1. — Flemish Money. — How much British money
is equal to 1054 guilders, 7 3tiy~s, the excliance bcine; 33*.
4d. Hemish to £1 British 1
S3.S. 4.,;. : r054 g. 7 st. : : £1 : ?
12 20
4U0 pence.
21087 stivers.
2
400)42174 Flemish pence.
_£10r,435 = £105 8s. 8iJ.
£1, the amount of British money considei'cd in the rate,
is put in the third term , 335. 4d.. its value in foreign money,
in the first; and 1054 g. 7 gt . the money to be reduced,
in the second.
9. How many pounds sterling in 1680 guilders, at
335. 3d. Flemish per pound sterling ,? A71S. JE168 8s.
10. Reduce 6048 guilders, to Rritisli money, at 33.?.
\\d. "Flemish per poun'i British .? Ans. i.'594 7a.
It T» 1
XI. jL'teuuco
money, at 34^.
£198 85. 61 f^^,
W.7
'Jit,
04S guilders, L. sUveis, to British
Flemish per pound sterling > Am
M'r
EXCHANGE.
375
lit. npw many pounds sterling in 1000 guilders, 10
stivers, exchange being at 335. 4d. per pbuud sterling ?
Jbis. iiilOO Is. *
Example 2. — Hamburgh Money. — How much British
money is equivalent to 476 marks, 9 skillings, the exchange
being 33s. iid. Flemish per pound British '?
s, d. m. 8.
33 6 : 476 9 : : £1 : I
12 32 2
ling >
16. Reduce
402 grotes. 15232f 19'=15251i grotes.
402 )152511
£379386=£37 I85. 9d.
Multiplying the schillings by 2, and the marks by 32,
reduaes both to pence.
13. How much British money is equivalent,to 3083
marks, 12 schillings Ilanibro', at 325. 4d. Flemish per
pound sterling .? Ans. £254 65. 8^^.
14. How much English money is equal to 5127 marks,
5 Schillings, Hambro' exchange, at 36s. 2d. Flemish
per pound sterling ? uins. i£378 Is.
15. How many pounds sterling in 244S marks, 9
schillings, Hambro', at 32^. 6d. Flemish per pound ster
A/is. £200 105.
7854 marks, 7 schillings Hambro*, to
British money, exchange at 345. lid. Flemish per
pound sterling, and agio at 21 per cent. ? Ans. £495
Ids. Old
Example 3. — French Money. — Reduce 8654 francs, 42
centimes, to British money, the exchange being 23f., 50c.,
per £1 British.
f. c. f. 0. 865442
23 50 : 8054 42 : : 1 : ■23:50==^368 5s. 5^.
42 centimes are 042 of a franc, since 100 centimes make
1 franc.
17. Reduce 17969 francs, 85 centimes, to British
money, at 23 franc \, 49 centimes per pound sterling >
Ans. £765.
18. Reduce 7672 francs, 50 centimes, to British
money, at 23 francs, 25 centimes per pound sterling ?
Ans. £330.
276
EXCHANGE.
10. EetlucG 1,5647 francs, 36 centimes, to British
money, at 23 francs, 15 centimes per pound sWlinf/ ?
A71.S. £675 ISs. 2ld. • ^ ^ o
20. lieduce 450 francs, 58^^ centimes, to British
money, at 25 francs, 5 centimes per pound sterling >
Ans. £nQ Us. ^ '
Example 4.
■ Pmiuguese
Money. —RovT much British
money is equa to 540 milrees, 420 rees, exchange beino; at
OS. m. per milree ? o o
m. m. r. s. d.
1 : 540420 : : 5 G : 540420x5s. 6c^.=£148 125. 3,^^.
^,V!".^^^^® *^^*^ S>^i*i»h money is the variable quantity,
and OS. M. is that amount of it which is considered in the
The rees are clianged into the decimal of a milree bv
putting them to the right hand side of the decimal point
since one reo is the thousandth of a milree.
21. In 850 milrees, 500 rees, how much British
money, at 55. 4d. per milree } Ans. £22Q \Qs.
22. Reduce 2060 milrees, 380 roes, to English money
at 55. Q^d. per milree } Ans. ^£573 0^. lOi^.
23. In 1785 milrees, 581 rees, how m*any pounds
sterling, exchange at 64i per milree.? Ans £479
175. Qd.
24. In 2000 milrees, at 5^. 81^^. per milree, how
many pounds sterling.? Ans. £570 165. S^^.
Example 5 —5;,_aH?5/i ilfonei/.— Reduce 84 piastres, 6 reals,
IJ maravedi, to British money, the exchange beino 4Ur/ the
piastre. ® o • ^""
r.
6
8
m.
19
d.
40
8
34
272
678 reals.
34
23052 maravedi.
49
272)1129548
41527, &c.=£17 %:. 02d.
EXCHANGE.
277
EXERCISES.
25. ReducG 2448 piastres to British money, exchange
at 50^. sterling per piastre ? Ans. i£;510.
26. Keduco 30000 piastres to British money, at 40d.
per piastre ? Ans. £5000.
27. Reduce 1025 piastres, 6 reals, 22if ^ maravedi, to
British money, at :^9ld. per piastre ? Ans. i2167 15s. 4d.
Example 6. — American Money. — Reduce 37G5 dollars to
British money, at 4s. per dollar. 4s.=£\ : therefore
5 )37G5 dol. del. s. £
753 is the required sum. Or 1 : 3765 : : 4 : 753
28. Reduce £292 3.?. 2^d. American, to British money,
at 66 per cent. .? Am. £176.
29. Reduce 5611 dollars, 42 cents., to British money,
at 4s. d^d. per dolLir ? Ans. £1250 175. 7d.
30. Reduce 2746 dollars, 30 cents., to British money,
at 45. S^d. per dollar ? Aqis. £589 Gs. 2^d.
From these examplcM the pupil will very easily under
stand how any other hind of foreign, may be changed
to British money.
8. To reduce Britisli to foreign money —
Rule. — Put that amount of foreign money which is
considered in the rati* of exchange as the thjrd term,
its value in British money as the first, and the British
money to be reduced as the second term.
ExAMPLK 1. — Flemk'i Money. — How many guilders, &o.,
in jC2oG 149. 2:1. Britisli, the exchange being 34s. 2d. Flemish
to £1 British I
£ £ s. d. s. d.
1 : 23G U 2 :: 34 2 : ?
20 20 12
20
12
240
4734
12
5G8UM.
410
410 pence.
24Qyr>202100
T2" )970504, &c.
20)8087 _ji_
iC404 7 U Flenieh.
N 2
278
EXCHANGE.
We might take parts for the 34.s. 2d.—
345. 2d.=£l 4 10s.44s.+2c/.
£> £ s. d.
^ei = 1 23G 14 2
I0s.= i 118 7 1
47 6 10
5i
i
4s.=a ^
2^^"=rL (aV of 1) 1 19
£404 7 61 Flemish.
EXERCISES.
31. In £100 l5., how much Flemish money, exchan<ro
at 335. 4d. per pound sterling? Am. 1000 guilders,
10 stivers. '
32. Reduce £168 85. dji^d. British into Flemish,
exchange being 335. 3d. Flemish per pound sterling ?
uItw. 1680 guilders. ^ "
33. In £199 ll5. 10^/j^.Britisli, how much Flemish
J!!?i!?^' exchange 345. 9^. per pound sterling ? Ans.
2080 guilders, 15 stivers.
34. Reduce £198 85. e^d. British to Flemish
inoney, exchange being 345. 5d. Flemish per pound
sterling > Ans. 2048 guilders, 15 stivers.
in £'9aT''p^7?""*T^^ JMovj^i/.How many marks, &c.,
in £24 65. British, exchange being 335. 2d. per £1 British l
£1
20
20
£24 65.
20
486
398
33s.
12
2d.
398 grotes.
20 )193428
2)9671 8 pence.
16 )4835 schillings, 1 penny.
302 marks, 3 schillings, 1 penny.
35. Reduce £254 65. 8d. English to Hamburgh
money, at 325. 4d. per pound sterling.? ]Ans. 3083
marks, 12 stivers.
36 Reduce £378 I5. to Hamburg money, at 365
2d. Flemish per pound sterling ? Ans. 5127 marks.
5 schillings. _ '
37. Keduce £536 to Hamburgh money, at 865. 4d
per pound sterling .? Ans. 7303 marks.
EXCHANGE.
279
38. Reduce JB495 155, OJ<Z. to Hamburg currency,
at 345. lid. per pound sterling ; agio at 21 per cent. ?
Ans. 7854 marks 7 schillings.
Example S. — French Moncij. — How much French money
is equal in value to £83 2s. 2d., exchange being 23 francs
25 centimes per £1 British ^
£ £ s. d. t
1 : 83 2 2 : : 2325 : ^
20 20
20 1662
12 12
240 19946
2325
240 )46374450
193227, or 19322f. 70c. is the required sum
39. Reduce £274 55. Oc?. British to francs, &c., ex
change at 23 francs 57 centimes per pound sterling r
Alls. 6464 francs 96 centimes.
40. In £765, how many francs, &c., at 23 franca
49 centimes per pound sterling } Ans. 17969 franca
85 centimes.
41. Reduce £330 to francs, &c., at 23 francs 25 cen
times per pound sterlijjg .? Ans. 7672 francs 50 cents.
42. Reduce £734 45. to French money, at 24 franca
1 centime per pound sterling } Ans. 1769 francs 42J
centimes.
Example 4. — Portuguese Money. — How many milrees anrf,
rees in £32 6s, British, exchange being 5s. 9(/. British pe
milree 1
s. d. £ s.
5 9 : 32 G : : 1000 : ?
12 20
69
646
12
7752
1000
69)7752000
,«quircd sura
112348 rce8=112 milrees 348 rees, is tno
260
XXCHANGE.
43. Reduce £226 16^. to milrecs, &c., at 5^. 4d. per
milrce ? Ans. 850 milrees 500 roes.
44. Reduce £^479 17*. 6d. to milrees, &c., at 6Ud
por milrec ? Ans. 1785 milrees 581 rees. ' *
45. Reduce £570 16.9. 8^. to milrees, &c., at 5*. Sid
per milree ? Ans. 2000 milrees.
46. Reduce £715 to milrees, &c., at 5*. 8d. permU
ree ? Ans. 2523 milrees 529/^ rees.
. ^^£^S^F,.5'Si'«ww;i Morm/.— Row many piastres, &o.,
in £02 British, exchange being 50d. per piastre '»
d. £
50 : 62 : : 1 : ?
20
1240 p. r. m.
12 ^97 32if , is the required sum.
50 )14880
^ 2976 piastres.
8
48 reals.
34
50)1632
32M maravedis.
•*7. How many piastres, &c., shall I receive for £510
sterling, exchange at 50c?. sterling per piastre ? Ans.
2448 piastres.
48. Reduce £5000 to piastres, at 40^. per piastre >
Ans. 30000 piastres.
49. Reduce £167 15*. 4d. to piastres, &c., at SO^d.
per piastre ? Ans. 1025 piastres, 6 reals, 22^4^ mara
vedis. '
^ 50,. Reduce £809 95 8d. to piastres, &c., at 40J. per
piastre ? Arts. 4767 piastres, 4 reals, 2yVV maravedis.
Example Q.— American Money .—Reduoe £176 British to
American currency, at 66 per cent.
£ £ £
100 : 176 :: 166 ; :
166
100)29216
£292 35. 2i(/., is the required sum.
EXCHANGE.
281
EXERCISES.
61. Reduce £753 to dollars, at 4s. per dollar > Ans.
3765 dollars.
52. Ileduce ^£532 4s. Sd. British to American money,
at 64 per cent. > Ans. £872 175. 3d.
53. Ikduce £1250 17s. 7d. sterling to dollars, at
4$. 5^d. per dollar ? Ans. 5611 dollars 42 cents.
54. Ileduce £589 6s. 2^%d. to dollars, at 4s. S^d.
per dollar } Ans. 2746 dollars 30 cents.
65. Reduce £437 British to American money, at 78
per cent. ? Ans. £777 17s. 2^d.
9. To reduce florins, &c., to pounds, &c., Flemish —
Rule. — Divide the florins by 6 for pounds, and —
adding the remainder (reduced to stivers) to the stivers
—divide the sum by 6, for skillings, and double the
remainder, for grotes.
Example.— How many pounds, skillings, and grotes, in
105 florins 19 stivers '?
f. St.
6)165 19
£21 13s. 2d., the required sura.
6 will go into 1G5, 27 times— leaving 3 florins, or 60 stivers,
"which, with 19, make 79 stivers ; 6 will go into 79, 13 times
leaving 1 5 twice 1 are 2.
10. Reason of ^he Rule.— There are 6 times as many
florins as pounds ; for we find by the table that 240 grotea
make £1, and that 40 C^*") grotes make I florin. There are
6 times as many stivers as skillings ; since 96 penniugs^make
1 skilling, and 16 (V) pfennings make one stiver. Also, sinca
2 grotes make one stiver, the remaining stivers are equal to
twice iiH many grotes.
Multiplying by 20 and 2 would reduce the florma to grotes ;
and dividing the grotes by 12 and 20 would reduce thorn to
pounds. Thus, using the same example—
f. St.
165 19
20
3319
2
12 )6638 
20)653_ 2
£27 ]3s. 2d., as before, is the result.
f '
?l
«
282
EXCHANGE.
EXERCISES.
56. Ill 142 florins 17 stivers, how many pounds, &c.,
Atis. £23 16*. 2d.
57. lu 72 florins 14 stivers, how many pounds, &c.,
Ans. £\2 2s. 4(1.
58. In 180 florins, how many pounds, &c. } Am. iE30
11. To reduce pounds, &o., to florins, &c. —
Rule. — Multiply the stivers by 6 ; add to the producfi
half the number of grotes, then for every 20 contained
in the sum carry 1, and set down what remains above
the twenties as stivers. Multiply the pounds by 6, and,
adding to the product what is to be carried from the
stivers, consider the sum as florins.
Example. — How many florins and stivers in 27 pounds,
13 skillinga, and 2 grotes ?
£f s. d.
' 27 13 2 .
6
165fl. 198t., the required sum.
6 times 13 are 78, which, with half the number (f ) of
grotes, make "^0 stivers — or 3 florins and 19 stivers (Z twenties,
and 19) ; putting down 19 we carry 3. 6 times 27 are 1G2,
and the 3 to be carried are 165 florins.
This rule is merely the converse of the last. It is evident
that multiplying by 20 and 12, and dividing the product by 2
and 20, would give the eamo result. Thus
£ s. d.
27 13 2
20
568
. 12
2)6638
20)3319
165fi. IDst, the same result as before.
EXERCISES.
59. How many florins and stivers in 30 pounds, 12
skillings, and 1 grote ? Ans. 183 fl., 12 st., 1 g.
60. How many florins, &c., in 129 pounds, 7 skil
linffs ? Ans. 776 fl. 2 st.
61. In 97 pounds, 8 skillings, 2 grotes, how many
florins, &c. : Ans. 584 fl. 9 st.
ARBITRATIOX OF EXCHANGES.
283
QUESTIONS.
1. What is exchange ? [1].
2. What is the difference between real and imagin
ary money ? [!]•
3. What are the par and course of exchange ? [IJ.
4. Wliat is agio? [3].
5. What is the difference between current or cash
noney and exchange or bank money ? [3] .
6. How is bank reduced to current money ? [4].
7. How is current reduced to bank money ? [6] .
8. How is foreign reduced to British money ? [7] .
9. How is British reduced to foreign money ? [8].^
10. How are florins, &o., reduced to pounds Flemish,
11. How are pounds Flemish, &c., reduced to florins,
fee? [11].
ARBITRATION OF EXCHANGES.
12. In the rule of exchange only two places are con
ecfned ; it may sometimes, however, be more beneficial
«o the merchant to draw through one or more other
places. The mode of estimating the value of the money
of any place, not drawn directly, but through one or
more other places, is called the arUtration of exchanges^^^
and is either simph or cortipound. It is " simple "
when there is only one intermediate place, " compound "
when there are 7/wre than one.
All questions in this rule may bu solved by one or
more proportions. ,
13. Simple Arbitration of Exchanges.— Given the
course of exchange between each of two places and
a thud, to find the par of exchange between the
former. , i • x
Rule.— Make the given sums of money belonging to
the third place the first and second terms of the propor
tion ; and put, as third term, the equivalent of what is
in the first. The fourth proportional will be the value
of what is in the second term, in the kind of mQuey
contained in the third term.
t
284
ARnrrnATioN of exchanges.
ExAMPLK. — If London cohanfoa with T'nrm nf in; .^
franc, an.l with Ani.tcnJa.u at 31.. njp Vi Z ul^ wCJ
ought to bo tho cour.s« of exoha,.:o,Tctt of p2 t;
Amstonhun that a n.erohant n.ay without loss ron^frca
L.Midon to Amsterdam through I'aria '? ""cirom
Df £lV'.\^n ■• ^"^' ^'/ ^*K^ equivalent, in Klomi«h money,
Fiefuilh mi!;:;'" "' '''■ ^^''^' ^^^ ^' ^ ^'^'^"^) ^^
^^^ • 240
'^Tl andlO f ^^'h?''f'^^ ?' «f ^ franc, in Flemish nu.ney.
that which belongs to the third place; and 34. 8 i/'tho
given equivalent of £1. ^ ' ^ Jii. oa. is tno
It is evident that, 17U. (Flemish) bein the value of in,/
<,i>iitisn;, out lie will not recp vf> ITJ^,; f.n. fi.,* i v,
EXERCISES.
is 1^5 o5' '^"^^^"Se between London and Amstordani
wh?l;!f'''^"^'' "^^t<^«dto Petorsburgh 5000 ruble. •
7'lt fo/"^:Tu''7^^^^^^ «nd London
ih c t oOd per ruble, but between Petcrsburrrh anl
Holland It IS at 90^. Plomish per ruble, and Holl d
Which will be the more advantageous method for Lon '
don be drawn uponthe direct°or the indirect ? Ans
Jjondon wdl .o;iin ]e9 n? 1 en^/ :p u ^
by way of Holland "» '' '^ '' ""*"^ P"^'""'"^
5000 rnblos— ^1041 T?. 17 p ... , r,,,
but ^1875 Fre«i;h=il^32 1 l^lVlufh.'" ^'^""^" '
ARBITRATION OV EXCHANGES.
385
14. Compound Arhitrntion of Exchanges. — To find
what should bo the course of exchange between two
places, through two or more others,, that it may be on a
par with the course of exchange between the same two
places, dircdly —
IluLE. — Having reduced monies of the same kind to
the same denomination, consider each course of exchan<»o
as a ratio ; set down the dift'orent ratios in a vertical
column, so that the antecedent of the second sliall be
of the same kind as the consequent of the first, and the
antecedent of the third, of the same kind as the conse
quent of the second — putting down a note of interroga
tion for the unknown term of the imperfect ratio. ThcL
divide the product of the consequents by the product of
the antecedents, and the quotient will be the value of tho
given sum if remitted through the intermediate places.
Compare with this its value as remitted by tho direct
exchange.
15. ExAMPLK.— £824 Flemish being due to me at Am
sterdam, it is remitted to France at IGrf. Flemish per franc;
from Franco to "Venice at 300 francs per GO ducats : from
Venice to Hamburgh at lOOtZ. per ducat ; from Hamburgh
to Lisbon at 50f/. per 400 rees ; and from Lisbon to England
at 5.S. 8^/. sterling per milrce. Shall I gain or lose, and how
much, tho exchange between England and Amsterdam being
34i'. 4t/. per XI sterling ?
\^d. : 1 franc.
300 francs.: GO ducats.
1 ducat : 100 pence Flemish.
50 pence Flemish : 400 roes.
1000 rees ; G8 pence BritisJi.
'? : £824 Flemish.
^XC0xl00x400x68x824 ,.^
'10X300X1X50X1000 =^'^ '^"^ ''^'^^'° *^^' **^^^"«
[Sec. V. 47]) 11^^=£5G0 Gs. A\d.
But the exchange between England and Amsterdam fd
£824 Flemish is £480 sterling.
Since 34s. M. : £824 : : £1 : .^^^'^.^£430.
I gain therefore by the circular
minus £480=£80 65. Aid.
34.S. 4d.
exchange X5G0 G*. 4u.
286
AKniTRATION OF RXCnANQES.
If commission ia chaxf^d in any of the places, it must
bo do(Juct(!d from tho value of tho sum which cuu bo
obtained in that place.
Tho procoss given for tho compound arbitration of ox
cliiin;5o may bo provo<l to bo correct, by putting down tbo
difFerent proportions, and nolving tbcm in Hueeeswion. 'Ibus,
if 10.'/. aro equal to 1 franc, what will 300 francs (=00
ducats) bo worth, ff tlio quantity last found is tho valuo of
00 ducats, what will be that of cue dueat (=3l00t/.), &o. '?
EXfiKCLSES.
3. If London would remit iDlOOO sterling to Spain,
tho direct exchange being 42),(l. per pia.stre of 272
maravedis ; it i.s ankoA whether it will bo more profit
able to remit directly, or to remit first to Holland at
3o5. per pound ; thence to France at Id^d. per franc ;
thence to Venice at 300 francs per 60 ducats ; and
thence to Spain at 3G0 maravedis per ducat ? Ans.
The circular exchange is more advantageous by 103
piastres, 3 reals, lOf^ maravcdLs.
4. A merchant at London has credit for 680 piastres
at Leghorn, for which ho can draw directly at oOd. per
pia.stre ; but choosing to try tho ciicular way, they aro
by his orders remitted first to Venice at 94 piastres per
100 ducats; thence to Cadiz at 320 maravedis per
ducat ; thence to Lisbon at 630 rces per piastre of 272
maravedis; thence to Amsterdam at 5 W. per crusade
of 400 rocs ; thence to Paris at IS^d. per franc ; and
thence to London at 10^^?. per franc ; how mucb is tho
circular emittanco better than tho direct draft, reckon
ing I per cent, for commission ? Ans. ^£14 12s. l^d
16. To estimate the gain or loss per cent. —
lluLE. — Say, as the par of exchange is to the c; urso
of exchange, so is iElOO to a fourth proportional. From
this subtract £100.
Example. — ^The par of exchange is found to be IS^d.
Flemi.sh, but tho cour.se of exchange is Idd. per fraiic ;
what is the gain per cent. ?
£19x100
lo ia.
fAOO
'M
— =X104 7*. Ud.
Thu.s I
X4 Is. IJ
If in
paid, it i
5. Th
but tho <
cent. ?
6. Th
course ik
6*. lli</
7. Th
course of
Ans. £1
1. W]
2. Wl
pound ai
3. AVI
4. Wl
5. He
any plac
6. Ho
17. T
gain or 1
certain ]
Given
gain or 1
KULE
and at tl
or loss
ExAMr
G(i., and i
Thetc
The tc
Thetc
I'UOFIT AND LOM.
287
Thus (ho piiu por ccnt.=,C104 7s. l^^ nilnuH £100=*
X4 7v. 11(/. if the merchant remits through I'liria.
It' in remitting through Paris oommisHiou must ba
paid, it is to be deducted from the gnin.
EXERCISES.
5. Tho par of exchange is found to bo \8^d. Flemish,
but the course of exchange is 19t/., whatis the gain per
cent. ? Ans. £4 ISs. 2</.
6. Tho par of exchange is 17 ^d. Flemish, but tho
course Is 18tZ., what is the gain per cent. ? Ans. £4
6s. UU.
7. The par of exchange is 18^^. Flemish, but tho
course of exchange is 17^rf., what is the loss per cent. ?
Ans. £1 165. 2d.
QUESTIONS.
1. What is meant by arbitration of exchanges .? [12].
2. What is the difference between simple and com
pound arbitration } [12].
3. AVhat is the rule for simple arbitration ? [13].
4. What is tho rule for compound aibitration ? [14].
5. How arc we to act if commission is charged m
any place .? [15].
6. How is the gain or loss per cent, estimated } [16].
PROFIT AND LOSS.
. 17. This rule enables us to discover how much we
gain or lose in mercantile transactions, when we sell at
certain prices.
Given the prime cost and selling price, to find the
gain or loss in a certain quantity.
KuLE. — Find the price of the goods at prime cost
and at the selling price ; the difference will be the gain
or loss on a given quantity
Example. — What do T gain, if I buy 460 lb of butter at
ijd.j and sell it at Id. per lb ?
The total prime cost is 460J.x6=2760f?.
Tlic total sGlliiig price is lOuCi.X i=o^^Od.
The total gain is o220(/. minus 27G0J.=460c/.=jCl 18s. id.
r.
i ♦
i i
288
PROFIT AND LOSS.
%
EXERCISES.
1. BougLt 140 ft) of butter, at lOd. per ih, and eold
it nt 14d. por ft) ; what was gained ? Ans. £,'1 6s. 8</.
2. Bought 5 cwt., 3 qrs., 14 lb of cheese, at £2 I2s.
per cwt., and sold it for d22 185. per cwt. What was
the gain upon the ^vholc ? Ans. £1 15s. 3d.
3. Bought 5 cwt., 3 qrs., 14 ft) of bacon, at 345. per
What was the
cwt.
and sold it at 365.
4d. per cwt.
gain on the whole .'' Ans. I3s. 8^d.
4. If a chest of tea, containing 144 ft) is bought
for 6s. 8ft. per ft), what is the gain, the price received
for the whole being £57 10s. } Ans. £9 lOs.
18. 1*0 find the gain or loss per cent. —
Rule.: — Say, as the cost is to the selling price, so Is
£100 to the required sum. The fourth proportional
minus £100 will be the gain per cent.
Example 1. — What do I gain per cent, if I buy 1460 lb
of beef at 3(Z., and sell it at Z^d. per Bb '^
3(Z.xl460=4380tf., ia the cost price.
And 3i(/.xl4G0=5110rf., is the selling nrice.
5110 X 100
Then 4380 : 5110 : : 100 : — ^^^ — = £116 13s. 4d.
Ans. £116 13.S. 4d. minus £100 (=£1^ 135. 4d.) is the gain
per cent.
REAijON OF THE RuLE. — The price is to the price plus the
gain in one case, as the price (£100) is to the price plus the
gain (£100fthe gain on £100) in anotiicr.
Or, the price is to the price plus the gain, as any multiple
or part of the former (£100 for instamse) is to an equimultiple
of the latter (£lOOfthe gain on £100).
Example 2. — A person sells a horse for £40, and loses 9
{)er cent., while he should have made 20 per cent. What ia
lis entire loss "?
£100 minus the loss, per cent., is 1o £100 as £40 (what
the horse cost, minus wliat ho lost by it) is to what it cost.
01 : 100 : : 40 : — — — =£43 19*. liJ., what the horse cost.
But the person should have gained 20 per cent., or ^
of the price j therefore his profit tihould have been
PROFIT AMI LOSS.
289
£ x. d.
3 19 l.V ia the difference between cost and selling price.
8 15 9^ is what he should have received above cost.
12 14 11} is his total loss.
so IS
, or \
been
EXERCISES.
5. Bought beef at 6(Z. per lb, and sold it at ^d.
What what was the gain per cent. } Ans. 331.
6. Bought tea for' 5s. per lb, and sold it for 3s.
What was the loss per cent. > Ans. 40.
7. If a pound of tea is bought for Qs. Qd.^ and sold
for Is. 4d.^ what is the gain per*cent. ? Ans. 12ff .
8. If 5 cwt., 3 qrs., 26 lb, are bought for £9 85.,
and sold for £11 185. 11^., how much is gained per
cent. } Ans. 27 ^V^.
9. When wine is bought at 175, Gd. per gallon, and
sold for 27.V. 6c/., what is the gain per cent. ? Ans. 57^.
10. Bought a quantity of goods for j£60, and sold
them for ^£75 ; what was the gain per cent. .'' Ans. 25.
^11. Bought a tun of wine for £50, ready money, and
sold it for £54 IO5., payable in 8 months. How much
per cent, per amium is gained by that rate .'' Ans. 13^.
12. Having sold 2 yards of cloth for II5. 6</., I
gained at the rate of 15 per cent. What would I have
gained if I had sold it for 12?. t Ans. 20 per cent.
13. If when I sell cloth at 75. per yard, 1 gain 10
per cent. ; wh t will I gain per cent, when it is sold for
85. 6i. .? Ans. £33 Us. 5^d.
'Is. : 8.S. 6(!. •: £110 : £133 lis. 5\d. And £133 II5.
5!/L— £100=£33 il.^ 5 i(Z., is the required gain.
19. Given the cost price and gain, to find the selling
price —
Rule. — Say, as £100 is to £100 plus the gain per
cent,, so, is the cost price to the required selling price.
Example. — At what price per yard must I sell 427 yards
of cloth which I bought for 19*'. per yard, so that I may
gain 8 per cent. 1
lOSxiO.N'.
100 : 108 : : 10s. : — iqq— =JC1 O5. G\d.
This result may be proved by the last rule.
290
rnOFIT AND LOSS.
EXERCISES.
14. Bought velvet at 4.?. 8f/. per yard ; at what price
must I sell it, so as to gaia 12^ per cent. ? Ans. At
55. 3d.
15. Bought muslin at 55. per yard ; how must it be
sold, that I may lose 10 per cent. ? Ans. At 4i*. 6d.
16. If a tun of brandy costs £40, how must it be
sold, to gain 6i per cent. ? Ans. For j£42 10a\
17. Bought hops at ii4 165. per cwt. ; at what rate
must they be sold, to lose 15 per cent. .? Ans. For £4
Is. l\d.
18. A merchant receives 180 casks of raisins, which
stand him in \Qs. each, and trucks them against other
merchandize at 28s. per cwt., by which he finds he has
gained 25 per cent. ; for what, on an average, did he sell
each cask ^ Ans. 80 lb, nearly.
20. Given the gain, or loss per cent., and the selling
price, to find the cost price —
Rule. — Say, as JGJIOO plus the gain (or as J3100 minus
the loss) is to £100, so is the selling to the cost price.
FiXAMPLB 1. — If I sell 72 K) of tea at (js. per lb, and gain
9 per cent., what did it cost per Jb ?
109 : 100 : : 6 : — Jq^=5s. M.
What produces £109 cost £100 ; therefore what pro
duces Os. must, at the same rate, cost bs. Qd.
Example 2. — A merchant buys 97 casks of butter at 30.«.
each, and selling the butter at £4 per cwt., makes 20 per
cent. ; for how much did he buy it per cwt. ?
30.v.x97=2910s,, is the total price.
Then 100 : 120 : : 2910 : ^~?^= 3492s., the
100
3492s.
Belling price. And ~q7)7' \='~£T^ )=43G5, is the number
of cwt. ; and ,jy=50]^* lb, is the uvcrage weight of each
cask.
lb lb .S. 110 vQ
Then 50}lj : 112 : : 30 : li"^'^'
: GO*. 8(/. = £3 65.
8(i., the required cost price, per cwt.
FELLOWSHIP.
291
EXERCISES.
,19i. Having sold 12 yards of cloth at 20*. per yard,
and lost 10 per cent., what was the prime cost? Ans.
22s. 2ld.
20. Having sold 12 yards of cloth at 20^. per yard,
and gained 10 per cent., what was the prime cost .'' Ans.
1 Si. 2fjd.
21. Having sold 12 yards of cloth for £5 14^., and
gained S per cent., what was the prime cost per yard.?
Ans. 8,?. 9§r/.
22. For what did I buy 3 cwt. of sugar, which I
sold for dE6 3a., and lost 4 per ceait. } Ans. For £Q
^s. IJ^.
23. For what did I buy 53 yards of cloth, which I
sold for £25, and gained £b \0s. per cent. } Ans. For
£23 135. 111(7.
QUESTIONS.
1. What is the object of the rule .? [17].
2. Given the prime cost and selling price, how is
the profit or loss found } [17].
3. How do we find the profit or loss per cent.? [18].
4. Grivcn the prime cost and gain, how is the selling
price found } [lJj.
5. Given the gain or loss per cent, and selling price,
how do we find the cost price .? [20] .
FELLOWSHIP.
21. This rule enables us, when two or more persona
aie joined in partnership, to estimate the amount of
profit or loss which belongs to the share of each.
h'idlowship is either single (simple) or double (com
pound). It is single, or simple fellowship, when tlia
diflerent stocks have been in trade for the same time.
It is double, or compound fillowsliip, when the difiercnt
^stoi'ks luive biieu employed for diJJV.reiU times.
This rule also enal)]es us to esti late how much of a
bankrupt's stock is to ])e given to each creditor.
293
FELLOWSHIP.
22. Single Felloivship. — Rulr. — Say, as the wliolo
stock^ is to the whole gain or loss, so is each prrson's
contribution ^o the gain or loss which belongs to him.
Example.— A put £720 into trade, B £340, and C
^eOGO ; and they gained Ml by the traffic. What is li'a
share of it ?
£
720 ..
340
960
2020 : £47 :: £310
X47X340
— 2020~~ ^^*
Each person's gain or loss must evidently be proportionai,
to his contribution.
EXERCISES.
1. B and C buy certain merchandizes, amounting
to £80, of whicli 13 pays £30, and deSO ; and they
gain £20. How is it to be divided .? Ans. B £7 10s ,
and £12 10.v.
2. B and C gain by trade £182 ; B put in £300,
and £400. What is the gain of each t Ans. B £78,
and C cii5l04.
3. 2 persons are to share £100 in the proportions
Of 2 to B and 1 to C. What is the share of each >
Am. B £66, C £33.
4. A merchant failing, owes to B £500, and to
£900; but has only £1100 to meet these demands.
How much should each creditor receive ? Atis. B £3924,
and C £707f ^'
5. Three merchants load a ship with butter; B
gives 200 casks, C 300, and D 400 ; but when they are
at sea it is found necessary to throAV 180 casks over
board. How much of this loss should fall to the share
of each merchant ? Ans.
60, and D SO.
6. Three persons are to pay a tax of £100 accord
ing to their estates. B's yearly prapcrty is £800, G'a
£600, and D's £400. How much is eacli person's share ?
B should lose 40 casks,
Ans.
n\.
is £44:^ C's £33^, and D's £223.
7. Divide 120 into throe sueh parts as shall be to
each other as 1, 2, and 3 ? Ans. 20, 40, and GO.
FELLOWSHIP.
293
S. A' ship worth £900 is entirely lost ; } of it be
Itmged to 13, J to C, and the rest to D. What should
be the loss of each, i3540 being received as insurance ?
Ans. B £45, G £90, and D £225.
9. Three persons have gained £1320 ; if B were to
take £6, C ought to take £4, and D £2. What is each
person's share ? Ans. B's £660, C's £440. and D's
£220.
10. B and C have gained £600 ; of this B is to
have 10 per cent, more than C. How much will each
receive .? Ans. B £314f , and C £2854.
11. Three merchants form a company; B puts in
£150, and C £260 ; D's share of £62, which they gained,
comes to £16. How much of the gain belongs to B,
and how much to C ; and what is D's share of the stock ?
Ans. B's profit is £16 165. 7j\d., C's £29 3s. 4^^d. ;
and D put in £142 12s. 2^^c?.
12. Three persons join ; B and C put in a certain
stock, and D puts in £1090 ; they gain £110, of which
B takes £35, and C £29. How much did B and C put
in ; and what is D's share of the gain ? Ans. B put
in £829 Gs. ll^J^., C £687 3s. 5iJ. ; and D's part of
the profit is £46.
13. Three farmers hold a farm in common ; one pays
£97 for his portion, another £79, and the third £100.
The county cess on the farm amounts to £34 ; what is
each person's share of it ? Ans. £11 18s. U^^d. ; £9
14s. 7^^d. ; and £12 6s. 4^^d.
23. Compound Fellowship. — Rule. — Multiply each
person's stock by the time during which it has been in
trade ; and say, as the sum of the products is to the whole
gain or loss, so is each person's product to his share of
the gain or loss.
KxAMPLK.— A contributes £30 for 6 months, B £84 for
11 months, and C £9G for 8 months; and they lose £14.
What is C's share of this loss 1
30 X 6=180 for one month. )
84x11=924 for one month. } =£1872 for one month.
y(3X 8=/U8 tor one mo
nth. V
1872 : £14 : : £708
£14x708
"1.S72
_ =£0 Is. 4ld., C's bharo
ir ; I
Ji^i
294
FELLOVVSiJip.
i
^„ ii iuunui , <inu, lor the same reason R'r no ^O'U
for Uio same time; ami C's m /"/«« oi =« ^ fu .^"*
EXERCISES.
in lio^'^i^""? '"5<^^^°t,s enter into partnership ; B puts
S is. /n % ^ ?r'^'' ? ^'^ '^' ^°r ^ months, Vu
1>^38 105 for 11 months; and they gain £86 16*
j^o lus., OS i.37 2s., and D's ^£24 4^
T^ in;«■^^^' '''"'^ ^P'J ^^ ^' *'^^ :^^^^'« ^^nt of a farm,
and 1) 50 for the rest of the time. How much of the
Ind D ^ii ^'''''' ^^^ ' ^''' ^ ^^^  ' ^ ^I^t't.
and In ^^^'^^^?"^t':^' A' I^' «"<i C, enter into partnership,
iAo wo" fVT "^^^^^SQl 13.. 4.Z.'^A's stock
C's i 2^ '?« ''^'.^ "'^l.^rl' ?'^' ^200' 3 months ; and
t s, X125, 16 months. What is each person's share of
131^47 '• ' '' '^^^' ^'' ^^0' ^^^ C'^ ^166
17. Three persons have received ^£665 interest B
^nvf 1^ *t ? ?' ^ ''^''''^^^ 5 ^^^^ "^"«^ is each person's
i)'si2oo ^''^' ^'^ ^^^^' ^'' ^^^^' ^^
trado* f'Zl^v^ '^I'^T ^«.°"ipa°y X's stock is in
trade 3 months and he claims J^ of the gain : Y's
Btock IS 9 months in trade ; and Z advanced^e756 for
4 months, and claims half the profit. How much did
X and Y contribute } Ans. X ^£168, and Y £280.
It follows that Y's gain was A. Then ' • » • • 4"T=.e.syA .
pay it60 ; the first sent into it 56 liorses for 12 days, tho
FELLOWSHIl'.
295
Bocond64 for 15 days, and tlio third SO for IS days.
What must each pay ? Am. The first must pay £17
10s,, the second £2o, and the third i;37 10a.
20. Three merchants are concerned in a steam vessel ;
the first, A, puts in £240 for 6 months ; the second, ]J,
a sum_ unknown for 12 months ; and the third, C, ^£160,
for a time not known when the accounts were settled. A
received £300 for his stock and profit, B £000 for his,
and C £200 for liis ; what was B's stock, and O's time ?
Ans. B's stock was £400 ; and C's time was 15 months.
If £300 arise from £240 in C months, £000 (B's stock and
profit) will bo found to arise from £400 (B's stock) in 12
months.
Then £400 : £160 :: £200 (the profit on £400 'n 12
montlis) : £80 (the profit on £100 in 12 months). And £l604
80 (£1G0 with its profit for 12 montlis) : £260 (£160 with
Its profit for some other time) :: J2 (the number of months
•^ *u s 260x12 ,
in the one case) : j^Xg^ (the number of months in the other
casc)=]3, the number of months required to produce the
difterence between £160, C's stock, and the £260, which he
received.
21. In the foregoing question A's gain was £60
during (3 months, li's £200 during 12 months, and C'a
£100 during 13 months; and the sum of the products
of their stocks and times is 8320. What wri(> their
stocks ? Ans. A's was £240, B's £400, and C's £160.
22. In the same question the sum of the stocks is
£800 ; A' stock was in trade 6 months, B's 12 months,
and C's 15 months; and at the settling of accounts,
A is paid £60 of the gain, B £200, and C £100.
What was each person's stock ? Ans. A's was £240,
B's £400, and C's £160. '
QUESTIOiS'S.
1. What is fellowship .? [21].
2. What is the difterence between single and douhle
fellowship ; and are those ever called })y any other
names .^ [21].
3. What are the rules for single, and double fellow
ship .? [22 and 23].
' litmmi
wSi
il
II
'l^H
m
296
BARTKU.
BAUTER.
25. Barter enables the merchant to exchange ono
commodity for another, without either loss or gain.
lluLE.— Find the price of the given quantity of ono
kind of merchandise to be bartered ; and then ascertain
how much of the other kuid tliis price ought to pux
. chase.
ExA>[PT,E 1.— How much tea, at 8s. per lb, ou^ht to be
t'ivcn for 3 cwt. of tallow, at £1 10s. Sd. per cwt. 1
£. s. d.
1 16 8
3
5
10 is the price of 3 cwt. of tallow.
And £5 10s.^8s.=13^, is the number of pounds of tea
which £o 10s., the price of the tallow, would purchase.
There must be so many pounds of tea, as will be equal to
the number of times that 8s. is contained in the price of tho
tallow.
E.vAMPi.E 2.— I desire to barter 96 lb of sugar, which
cost me Sd. per lb, but which I sell at 13rf., giving 9
months' credit, for calico which another merchant sells for
lid. per yard, giving months' credit. How much calico
ouglit 1 to receive l
I first find at what price I could sell my sugar, were I to
give the same credit as he does —
If 9 months give me 5d. profit, what ought 6 months to
giveT
9 :
5 .6X5 30 ._gv/
9 ~9~" ' ■
Hence, were I to give months' credit, I should charge
ll»f/. per lb. Next—
As my selling price is to my buying price, so ought his
soiling to be to his buying price, both giving the same credit.
lit : 8 :: 17 :5>^=12.Z.
Tlie ]irico oi my f?ugar, inuroiuri;, is t?o a <•"•> "'• '^^^•■•f
md of his calico, 12r/. per yard.
Hence "^^^=04, is tho required number of yards.
BARTER
297
EXERCISES.
1 . A mevcliant lias 1200 stones of tallow, at 2s. 3ld.
Iho Ktonc ; 13 has 110 tanned hides, weight 3994 lb, at
b^d. the lb ; and thoy barter at these rates. How much
ijwney is A to receive of li, along with the hides > Ans.
£40 ll5. 2hL , , ,^ ^j
3. A has silk at Ms. per !b ; B has cloth at 12s. 6rf.
which cost only 10s. the yard. How much must A charge
for his silk, to make his profit equal to that of B ? Ans.
17s. 6d.
3. A has coffee which he barters at lO^Z. the lb more
than it cost him, against tea which stands ]5 in lOs.,
but which he rates at 12s. Qd. per.tb. How much did
the coffee cost at first ? Ans. 3s. 4d.
4. K and L barter. K has cloth worth 8s. the yard,
which he barters at 9s. Sd. with L, for linen cloth at
3s. per yard, which is worth only 2s. 7d. Who has the
advantage ; and how much linen does L give to K, for
70 yards of his cloth .? Ans. L gives K 215f yards ;
and L has the advantage.
f). 1) has five tons of butter, at £2o lOs. per ton, and
lOi tons of tallow, at £33 15s. per ton, which he barters
witli ; agreeing to receive i2150 Is. 6d. in ready
money, and the rest in beef, at 21s. per barrel. How
many barrels 's he to receive > Ans. 316.
6. I hi've cloth at Sd. the yard, and in barter charge
for it at 13^/., and give 9 months' time for payment;
mo,. ''^ant has goods which cost him 12^. per
lb, an hich he gives 6 months' time for payment.
IIow hi : he charge his goods to make an equal
barter .^^ ...... At 17^^. ,^ , . .
7. I barter goods which cost 8d. per lb, but tor
which I charge 13f^., giving 9 months' time, for goods
which are charged at 17 d., and with which 6 months'
time are given. Required the cost of what I receive >
Ans. I2d.
8. Two persons barter ; A has sugar at Sd. per lb,
charges it at 13d., and gives 9 months time ; B has
at 12d. per lb, and charges it at 1 7d. per lb. How
time must B give, to make the barter equal?
6 months.
•f
298
ALLIGATION.
QUESTIONS.
1. What is barter ? [25].
2. Wiiat i.s tlie rule for k
'arter? [25],
ALLIGATION.
t IS called alligation medial; or what inredients wl
oe rcQuircd to np/w]i.«,> „ ^ • . ^^o^^uiLms will
tlioy will produce— 'fertaicnts, to Iind the mixture
nmnbc, of tl,e lo«.o.st denomination confined in th^
whole ,,„a„„ty, „„d tho qnotient will boX Ate or
d. d.
9X08 = 882
6x87 = 435
6x34 = 204 .
219 219)"l52l
Ans. Id. per ib, nearly.
The price of each e']"nr. is fhp nnmV.«« „*•
multipliod by tho iunn%;r if pou d ami fh^' ^'' P^"?^
whole is the mm of tho pricon B t 'if ° IQ ih /'"'"^ "^ ^^'°
cost lo21./., ono 11.. ov the '^1o[i, !^A, .^^?.^^*'^^"8'^^' ^^'ive
21UtI
part of thiH, li
I piirt of ]621t/ or 'A^'</ ~ ;
lust cost the
ALLIGATION.
299
KxAMPLK 2.— What will bo tho price iinv II) of a mixtiiro
ooiitainins !) lb G oz. of ten at 5s. Or/, per lb, 18 lb at (5>
per lb, and 4() lb 3 oz. at U.s. 4^^/. per lb «
lb oz. n.
9 6 at 5
18 G
46 3 9
d. £ s,
6 per lb= 2 11
per lb= 5 8
4iperlb=21 13
d.
GJ
9
1177 )29 12 G ;
Ans. 6d. per oz. nearly
73
IG
Il77 ouncoa.
And Gd. X 10=8.^., is the price per pound.
In this case, tho lowest denomination beinff outice.M wo
reduce the whole to ounces ; and having found the price of an
ounce, wo multiply it by IG, to find that of a pound.
E.YAMPi E 3.— A goldsmith has 3 lb of p;old 22 carats line,
and 2 lb 21 carats lino. What will bo tho linoutss of tha
mixture ?
In this case the value of each kind of inrrcdient is iT.nc
scnted by a number of carats —
lbs
3x22 = GG
2x21 = 42
5
5)108
Tlie mixture is^^ carats fine.
EXERCISES.
1. A vintner mixed 2 gallons of wine, at lis. por
gallon, with 1 gallon at 124., 2 gallons at 9^., and 4
gallons at 85. What is one gallon of tho mixture worth ?
Ans. 10s.
2. 17 gallons of ale, at 9d. per gallon, 14 at 7i^., 5
at 91^/., and 21 at 4ir/., are mixed together. How
much per gallon is the mixture worth ? Ans. 7j\d.
3. Having melted together 7 o?.. of gold 22^ carats
fine, 121 oz. 21 carats fi'no, and 17 oz. 19 carats fine, I
wish to know the fineness of each ounce of the mixture ?
Ans. 20f carats.
28. Alligation Alternafe. —Given the nature of the
mixture, and of the ingredients, to find the relative
amounts of the latter —
^ KuLE. — Put down the quantities greater than tho
given mean (each of them connected with the differenco
r •
300
AM.IOATION
between it and the moan, by tlio Higii — ) in one column ;
put tlio difforences botwcnn the remaining (luautitiea
and the moan (eonncctcd with the quantities to which
they belong, by the sign + ) in a column to the right
hand of the former. Unite, by a line, amlipliis with souio
viinus difference ; and then each difference will cxprii.ss
how much of the quantity, with whoso difference it is
connected, should be taken to form the required mixture.
If any difference is connected with more than one
other difference, it is to be considered as repeated for
each of the differences with which it is connected ; and
thef sum of the differences with which it is connected is
to be taken as the required amount of the (Quantity
whose difference it is.
Example 1.— How many pounds of tea, at 5.5. and 8.?. per
lb, would form a mixture worth 7."?, per tb '?
Price. Diflerences. Price, i
1
S. S.
The mean=8— 1
.V. s.
2f5=:thc moan.
1 IS connected with 2s., the difference l)otween the mean
and 5s. ; hence there must bo 1 lb at 5s. 2 is connocled
with 1, tlie difference between 8.<?. and the moan ; honco there
must be 2 lb at 8s. Then 1 lb of tea at 5s. and 2 ib at 8.s.
per ib, will form a mixture worth 7s. per lb — as may bo
proved by the last rule.
It is evident that any equimultiples of these quantities
would answer equally well ; hence a great number of answers
may be given to such a question.
Example 2.— How much sugar at Od, Id, 5d., and 10'/ ,
will produce sugar at 8>;/. per ib ?
Prices. Hirt'eronces. Prices.
The mean=
d. d.
91
102
d. d
3+5
the mean.
1 is connected with 1, the difference between Id. and the
mean ; hence there is to be 1 ib of sugar at Id. per lb. 2 is
connected with 3, the difference between 5d. and the mean ;
hcwee there is to be 2 lb at M.. 1 is connected with 1, the
difTerence between 9 J. and the mean ; hence there is to be
1 lb at 9f/. And 3 is connected with 2, the difference between
lOf/. and the mean; hence there are to bo 3 lb at ]0c/.
per ib.
AI.MUAMON.
301
CoDHcrniontly wo nro to tiiko I lb ut 7>l., and 2 lb at 5</.,
1 tb at ','*/., uud 3 11) ut lOil. If wo exuiuino wliat inixturo
tUeso will give [27], wo Hhall find it to bo tlio givon moan.
ExAMJM.K 3.— What quantities of tea at 4s., 6a., Ss.
0.«. por lb, will pi'oduoo aniixturo worth Ss. ?
I'lices. Dift'ureucei. Tricui,
S. S.
1 f4=tho mean.
and
Tho moan=
94
3, 1, and 4 aro connected with 1.?., the difforooo between
4<. juid tho mean ; thorofore wo aro to tako 3 lb f 1 lb } 4
lb of tea, at 4s. per lb. 1 ia connected with 3.'?., l.j,, and 4.s'.,
tho ditferoncos between 8,s\, Gs., and 9s., and tho moan,
thoroforo wo aro to take 1 lb of tea at 8s., 1 tb of toa at Ga.,
and I lb of tea at 9s. por lb.
Wo Und in this oxampio that 8s., 6s., and 9s. aro all oon
nocted with the same 1 j this shows that 1 lb of oaoh will
be required. 4s. is oonnoctod with 3, 1, and 4; there nmsU
bo, therefore, 3fl+4 lb of tea at 4s.
ExAMPMc 4. — How much of anything, at 3s., 4s., 5s., 7?.,
8s., 9s., lis., and 12s. por lb, would form a mixture worth
Gs per lb '?
Pricci. Diirorences. Prlcei,
Gs
1 lb at 3r, 2 lb at 4s., 3 lb at 7s., 2 lb at 8s., 35+6 (14)
lb at Ss., 1 lb at 9s., 1 lb at Us., and 1 lb at 12s. per lb, will
form tho required mixture.
29. Reason ok , the IIule. — The excess of one ingredient
above the mean is made to counterbalance what the other
wants of being equal to tho mean. Thus in example 1, 1 lb
at 5s. per lb gives a deficiency of 2s. : but this is corrected by
2s. excess in the 2 lb at 8s. per lb.
In example 2, 1 lb at Id. gives a deficiency of Id., 1 lb at 9^/.
gives an exce.ss of Irf. ; but the excess of Id. and the deficiency
of Id. exactly neutralize each other.
Again, it is evident that 2 lb at 5.Z. and 8 lb at 10^. are
Worth just as much as 6 lb at 8rf.— that is, Sd. will b« tha
ftverugo price if w« mix 2 ib afc iiU. with '6 lb at lOd.
302
ALLIGATIOiV.
■, •»«» #<i^
EXERCISES.
4. How much wine at 8s. 6d. and ds. per gallon will
make a mixture worth 8s. lOd. per gallon.? Ans. 2
gallons at Ss. 6d., and 4 gallons at Qs. per gallon.
5. IIow much tea at 65. and at 3s. Sd. per Il>, will
make a mixture worth 4s. Ad. per lb .? Am. 8 Sb at
(is.y and 20 lb at 3s. 8^. per lb.
6. A merchant has sugar at 5r/., 10^., and I2d. per
lb. How much of each kind, mixed together, wili be
worth M. per lb .? Am. 6 lb at 5^., 3 lb at IQd., and
3 lb at I2d. '
7. A merchant has sugar at bd., 10^., 12^., and 16'^
per lb. How many lb of each will form a mixture worth
lU. per lb? Am. 5 lb at bd., 1 lb at 10^., 1 lb at
12(Z., and 6 lb at 16^.
8. A grocer has sugar at bd., Id., 12d., and 13d..
per K). ; How much of each kind will form a mixture
worth lOd. per lb .? Am. 3 lb at 5d., 2 lb at 7d.,3fb
at 12d.j and 5 lb at 13^.
30. When a given amount of the mixture is required,
to find the corresponding amounts of the ingredients—
Rule.— Find the amount of each ingredient by the
last rule. ^ Then add the amounts together, and say, as
their sum is to the amount of any one of them, so is the
required quantity of the mixture to the correspondinff
amount of that one.
Example 1.— What must be the amount of tea at 4s. per
ft, m 736 lb of a mixture worth 5s. per lb, and containing
tea at bs., 8s., and l)s. per lb ?
To produce a mixture worth 5s. per lb, we require 8 lb
at 4s., 1 at 8s., 1 at 6s., and 1 at 9s. per lb. [28]. But all
ot these, added together, will make 11 lb*, in which there
are 8 lb at 4s. Therefore
lb
8x736 tt> oz.
=526 4y*y, the. required quantity
lb
11
lb
8
m
736
11
of tea at 4s.
That is. in 736 lb of flm mlx+nro thp""
m. at 'rs. per lb. The amount of each of the other ingre
dicnts may be found in the same way.
Triix tJ\^ xju\J lU I,",
ALLIGATIOX.
303
rrnJi^l^ F 'u}'T' ^^"^ of Syvacu.o, £;ave a certain
quaali y of p.ld o fom a ciwn; but when he received it,
suHpectmo; that th« goLl.niith had taken son^e of the gold
and «upp hed it« place by a b.ser metal, lie co,nmi«simied
Auh .).ed_(^S the celebrated mathematician of Syracuse, to
TZ'T^'x "' '^fP'^i'^"/^^^^^ ^vell founded, aril to what
oxtrnt Archiriiedes was tor some time unsuccessful in his
resoarches, unti one day, goin iuto a bath, he rcmark.Kl
that he displaced a quantity of water equal to his own bulk
Seeing at once that the same weight of different bodies
wou.d, Jf "nmcr,sod in water, displace very dilFcrent quan
tities of the fluid he exclaimed with delight that he had
found the desired solution of the problem Taking a mass
j>t guld equal HI weight to whatwa^ given to the gohlsmith,
he tound thnt it displaced less water than the crown : which
t.ieretoro, was made of a lighter, becnn.so a more bulky
mortal— and, consequently, was an alloij of goM 
jNow supposing copper to have boen the substance with
wliich the crown was adulterated, to find its amount
J.et the goiu given by Hiero have ww£rhed 1 lb, this
won d displace about O.IL' lb of water; 1 lb ^.f copper void
d.sp ace about 1124 3b of Avater; but let the criwn have
displaced only 072 it). Then "^ uo\mi .ia\e
(rold differs from 072, the memi, by— •020
Copper differs from it by . . fOlOl",
,T ., Copper. Di'I'LTPiices. (Jold.
Hence, ths moan=.=. 1124 0404 •020f052=thc mean.
Therefore 020 lb of copper and 0404 ib of gold would
t>r()duce the alloy in the crown. ^
l>ut the crown was supposed to weigh 1 ib ; therefore
•0G04 lb (020+ 0404) : 0404 lb • • lib • li^Mil"*
•0G04
•GG9=331 lb is
GC9 ib,_ the quantity of gold. And 1
thc quantity of coppc
EXERCISKS.
U. A diusrgist IS desirous of producinir, from medicine
at '>J'., (^'.v., S.v and 9.. per 3b, li cwt. of a mixture
worth 7s per ]b. How much of each kind must he
use for the purpose r Ans. 28 lb at 5.$., 56 lb at 6s.,
ufv xKf ai, 05., ana 2:i ih at bi'. per ib.
10. 27 lb of a mixture worth 4s. 4d. per Ib are re
qiured. It IS to contain tea at 5^. and at 3s. 6d. per
304
ALLIGATION.
lb. How mucli of each must bo used ? Ans. 15 ft) at
5i., and 12 ib at 3.9. 6cL
11. How much sugar, at Ad., Gd., and Sd. per ]b,
must there be in 1 cwt. of a mixture worth 7d. per ib }
Ans. 18 lb at 4(/., ISf lb at 6d., and 74 lb at 8d.
per lb.
12. How much brandy at 123., 135., 145. , and 14a'.
Gd. per gallon, must there be in one hogshead of a mix
ture worth 135. Gd. per gallon > Ans. 18 gals, at 125.,
9 gals, at 135., 9 gals, at 145., and 27 gals, at 145. Gd.
per gallon.
31. When the amount of one ingredient is given, to
find that of any other —
lluLE. — Say, as the amount of one ingredient (found
by the rule) is to the^iim amount of the same ingredient,
so is the amount of any other ingredient (found by the
rule) to the required quantity of "that other.
Example 1.— 29 lb of tea at As. per lb i« to bo mixed with
teas at (js., 85., and Ds. per lb, so as to produce what will be
•Vortli 5.s\ per lb. What quantities must be used '?
8 Ih of tea at 45., and 1 ib at 6s., 1 lb at 8s., and 1 lb at
9s., will make a mixture worth 5s. per lb [271. Therefore
8 ib (the quantity of tea at 4s. per Ib, as found by the rule) .
29 R) (the given quantity of the same tea) : : 1 lb (the
quantity of tea at Gs. per ib, as found by the rule) : 221^ ^^
8
rthe quantity of tea at 6s., Avhich corresponds with 29 lb at
4s. per lb) ==3^ lb.
We may in the same manner find what quantities of tea nt
8s. and 9s. per lb correspond with 29 lb— or i\\Q given amount
of tea at 4.s. per lb.
Example 2.— A refiner has 10 ov.. of gold 20 carats fine
and melts it with 16 oz. 18 carats fine. What must be
added to make the mixture 22 carais fine ?
10 oz. of 20 carats fine=10x20 = 200 carats.
16 oz. of 18 caratii fme=16xl8 = 288
26 : 1 : : 488 : 18}'[ carats, the
fineness of the mixture.
24 — 22=2 carats baser metal, in a mixture 22 carats fine.
24 — 18f=5j% carats baser metal, in a mixture 18 JH
carats fine.
Then 2 carats : 22 carats : : 5^^^ : 57 j''^ carats of pure
ALLIOATION.
305
fro] (1 required to ohanse 5 ■', carats baser metal, into a
mixture 22 carats line. Ikit tliero are already in the mixtura
1S:; Ciirats gcl.l; therefore 57^^,— 18j!;:=:i8f! carats ir<M
are to, l)e added to every ounce. There are 20 oz.; therefore
2GXoH.;=1008 carats of gold are wanting. There are
L4 carats ^( page 5) in^everyoz. ; therefore 'i;^^ caratsrr^12
'" """'" ' ' " ' There will then' he a uiixturo
oz. of gold must l)e added
containing
oz. car.
10X20
]()Xl8
42x24
car.
2')0
288
1008
08 : 1 oz. : : 14DG : 22 carats, the required finoness.
EXERCISES.
13. How iTiiicli tea at 6s. per lb must be tnixod with
12 ii) at 3i. ikl. per il), so that the mixture maybe
worth •].?. 4d. per lb .? Ans. 4f lb.
14. How much brass, at I4d. per tb, and pewter, at
lO^d. per lb, must I melt with 50 lb of copper, at 16V/.
per lb, so as to make the mixture worth Is. per lb ?
Ans. 50 lb of bra.s.s, and 200 lb of pewter.
15. How murdi gold of 21 and 23 carats fine must
be mixed with 30 oz. of 20 carats fine, so that the mix
ture uuiy bo 22 carats fine r A)is. 30 of 21, and 90
of 23.
16. How much wine at 7s. r^d.^ at 5.?. 2d., and at
4s. 2d. per gallon, must be miyxMl with 20 gallons at
O.v. 8^/. per gallon, to make the mixture worth 6s. per
gallon r Ans. 44 gallons at 7s. ixL, 16 gallons at oa
2d., and 34 gallons at 4i. 2d.
QUESTIONS.
1. What is alligation medial .? [26].
2. What is th,^ rule for alligation me lial > [27].
3. What is alligjition altoniato : [26 K
4. Whnt is the rule for Jilligatim alternate } [28].
5. What is the rule, v.hon a certain amount of t) <j
mixture is required .? [30] .
6. AVhat is the rule, when i\\(^. a; m\\\ 0\ C'l* or moro
of the ingredients is <^iveu .^ [31].
306
SECTION IX.
INVOr.UTION AND EVOLUTION, kc.
1. iNVOLUxroN. — A qnantlty wliicli is the product of
two or more factors, each of theiu llie same number, is
termed a power of that number ; and the number, mul
tiplied by itself, is said to })0 invclccd. Thus SXoXo
(:^125) is a " power of 5 ;" and 125, is 5 " hivolved."
A power obtains its denomination from the number of
times the root (or quantity involved) Is taken as a factor.
Thus 25 (=5X5) is tlie secovd power of 5. — Tlie
second power of any number is also called its square. ;
because a square surface, one of M'hose sid' s is expressed
by the given number, will have its area indicated by the
second power of that nun ber ; thus a square, 5 inches
every way, will contain 25 (the S(uare of 5) square
inches ; a Sfjuare 5 feet (svery way, will contain 25
.square foot, &c. 216 (6X0X<)) is "the lliird power of
6. — The third power of any nundjer is also termed its
mill ; because a cube, the length of one of avIi ^e sides
is expressed by the given number, will have ils solid
contents indicated by the third power e.f that number.
Thus a cube 6 inches every way, will contain 125 (the
cube of 5) cubic, or solid inches; a cube 5 feet every
way, will contain 125 cubic feet, tic.
2. In place of setting dov/n all the factors, we put
down only one of them, and mark how often they are
supposed to be set down by a small figure, which, since
it poin/s out the number of the factors, is called the
i7idc.x, or cxpinnanf:. Tlius ^^ is the abbreviation for
5x5 : — and 2 is th>5 index. 5^ moans 5X5X5X5X5,
or 5 in the fifth power S"* means 3X3X3X3, or 3 in
the fourth power. S' moans 8X8X8X8X^X8X8,
or 8 in the seventh power, &c.
3. Someti)nes the vinculum [See. IT. 5] is used in con
junction with the index ; thus 5f'82 means that the sum
of 5 and 8 is to be raised to the second power — this
INTOLUTION.
307
is very ciIiTerent from 5 ^+8 ° , wlncli means tlic sum of
the squares of 5 and S : 5 + 8= being 169 ; while 5^ + S''
is only 89.
4. Iq multiplication the multiplier may be considered
as a species of index. Thus in 187x5, 5 points out
how often 187 should be set down as an addend ; and
187X5 is merely an abbreviation for 187+187+187 +
187+187 [Sec. 11.41]. In 187% 5 points out how
often 187 should be set down as a factor ; and 187* ig
an abbreviation for 187X 187X 187X 187x 187 :— that
is, the " multiplier" tells the number of the addends^ and
the " index" or " exponent," the number of the factors.
5. To raise a number to any power —
Rule. — Find the product of so many factors as the
index of the proposed power contains units — each of the
factors being the number which is to be involved.
Example 1. — What is the 5th power of 7 "?
7» =7x7x7x7x7=10807.
Example 2. — What is the amount of £1 afc compound
interest, for 6 years, allowing G per cent, per annum 1
The amount of XI for G years, at 6 per cent, is —
_10GxlOGxlOGxl06xlOGxl06 [Sec. VII. 20], or
100"=141852.
We, as already mentioned [Sec. VII. 23], may abridge
<\.o process, by using one or more of the products, already
obtained, as factors.
■^ EXERCISES.
1. 3'=243.
2. 20'"=I0240000000000.
3. 3^=2187.
4. 105''=1340095r>40r)25.
5. 105''=l340095610G25.
^6. To raise a fraction to any power — '
Rule. — Raise both numerator and denominator to
that power.
Example. — (f)=^
to maliiply it ny itself. But to
multiply it by itself any nuuil)er of tiinos, we must multiply
its numerator by itwelf, and also its deuomiuator by itself, ihaf
number of times [Sec. IV. 00].
^os
EVOLUTION.
R /•.■^^7 'J Mil
"• U ^ — 11T38V 
() / o\n :i I :.';■.
7. To raise a mixed nunibor to any power —
lluLE. — llcduoo it to an improper fraction [Sec. IV
24] ; and then proceed as directed by the lust rple.
EXAMI'LK.— (21)4=()4=fyi^5.
EXERCISKS.
10 Kr.J  T.^
11. (3^)^=«u^;;^^
  ■■  .2 2 1 (iir
£9
8. Evolution is a process exactly opposite to mvolution ,
since, by means of it, v/e find what number, raised to a
given power, would produce a given quantity — the num
ber so fnund is termed a root. Thus wc " evolve " 25
when wo take, for instance, its square root ; that is, when
wc find what number, multiplied by itself, will produce
25, Roots, also, are expressed by e.rjjonenls — but as these
exponents are fractions, the roots are called ^^ fractional
powers." Thus 4^ means the square root of 4 ; 4^ the
cube root of 4 ; and 4^ tlie seventh root of the fifth power
of 4. Hoots are also expressed by ^, called the radical
sign. When used alone, it means the square root — thus
^3, is the square root of 3 ; but other roots are indicated
by a small figure placed within it — thus ^6 ; which
means the cube root of 5. ^7^ (7^)? is the cube root
of the square of 7.
9. The fractional exponent, and radical sign are some
times used in conjunction with the vinculum. Thua
4—3% is the s quare root of the difierence between 4
and 3 ; ^o{7^ or 5+7'^, is the cube root of the sura
of 5 and 7.
iO. To find the square root of any number —
Rule — I. Point off the digits in pairs, by dots ; put
ting one dot over the units' jo/acfi, and then another dot
over every second digit both to tha right and left of
the units' place — if there are digits at both sides of the
decimal point.
EVOLUTION.
309
IT. Find the highest immber the square of which
will not exceed the amount of the highest period, or
that which is at tlie extreuio hift— this number will bo
the first digit in tho required square root. Subtract its
square from the highest period, and to the remainder,
considered as hundreds, add the next period.
III. Find the highest digit, wliich being multiplied
into twice the part of the root already found (consi
dered as so many tens) , and into itself, tho sum of tho
products will not exceed the s^tm. of the last remainder
and tho period added to it. Put this digit in the root
after the one last found, and subtract the former si07>i
from the latter.
IV. To the remainder, last obtained, bring down
another period, and proceed as before. Continue this
process until the exact square root, or a sufiicicntly
noar approximation to it is obtained.
11. I'LxAMPLK.— What is the square root of 22420225 '^
22420225(4735, is the required root.
1G__
87)042 ,;;■ :"'•■ 
GOO ,,. . . . •
943)3302 ■ "^^ ' • .'■
2820 ..■■■ .. •
0405)47325
47325
22 i« tlie highest period; and 4^ is the highest square wlucli
doo.s not exceed it— we put 4 in tlie root, and subtract 4'',
or 10 from 22. This leaves 0, which, along with 42, the next
poriod. malccs 042.
We subtract 87 (twice 4 tcns{7, the highest digit yhicIi
wo can now put in the root) X 7 from 042. This loaves
33, which, along witli 02, the next period, makes 3302.
We subtract 043 (twice 47 tens \'i, the next digit of th(^
root) X3 from 3302. This lca.vo.s 473, ^^jiich, ixhmu V;:t'.^
25, the only remaining period, makes 47325,
We subtract 0405 (twico 473 tons J..'",, the np:.c digit of
the root) X5. Thi.s leaves n'^ romaiuder,
The given numbp,v, therefore, is exactly r. square; and
its squi\re root is 4735,
12. llKAsoiv OF I.— Wc point off ';„o .^ipitg of tlie given
square in pairs, and consider tlio ^^j,i^{)cr of dots as indicating
310
iJVOLUTIOr*
^^ii^ir^ ' "*■ ''■'='" '■" .""' '■'>"'• ''"'" ""Mor one nor two
the root—since it will be necessarv fn K^,^~\5 ^ '^ .^ ,
for each new digit; but Zr"o1hri^e*"w^rn«„?°b:?e°,K™''
Keabon or II —We subtract from llie Wcliost Mrio,l of '(!,.
fml 00?^"'°''.,""' '■'8''''" 'I""' "W«h £ nJt ™co°d U
dit of I „"■ "'? T' "' *■' "1""° »« the 8rst or hTlc,;
600 m JZ\, 'i?.'^"""^ by to digits mi. for iustance,"nto
will contain not only IO2 and 42 hnf nian +!;• li ^^^"^^^
rvf in oTiri .1 \u "l/ , , * ' "^^ *'So twice the product
£Xf ^r r s .t. rsriort^^i'triai
cedS it "WX'Z^ " ''^ "!° "«' »f tl.e root wl^iSp? .'
whenwesubracfS7v7 """Pf "''f'' "'•«*■■"*« the rule.
4000 =16000000 '
6420225
2X4000X700+700^= 6090000
2X4000X30+2X700X30+30*= 282900
8X4000X5+2X700X5+2X30X5+5^=17325
EVOLUTION
3tl
of twice tlic sum) of tlio parta of tlio mot nlrc.idy found,
jnulMphed by tlio ncAV digit, Tims 22420225, the 8quavo of
4785 contains 4000^f70030^f5^ and also Uvica 4000X
700 + twice 4000X30 4 twice 4000x5; plus twice 700x304
twice /00X6; phis twice 80x5:— that is. the square of each
ot Its parts, with the euui of twice tlio product of every two of
them (which is the same as each of tliem multiplied by twice
the sum of all the rest). This would, on examination, be
lound the case with the square e)f any other number.
If we examine the cxamjile given, we shall find that it will
not be necessary to bring down more than one period at a
time, nor to add cyphers to tlie quantities subtracted.
13. When the given square contains decimals —
; If any of the periods consist of decimals, the digits
m the root obtained on bringing down these periods to
the remainders will also be decimals. Thus, taking the
example jus t given, bu t altering th e decimal point, wo
Bh all have ^2 2420225= 473 5; V224 20225=4735.
^2242022 5 = 4735; V^2420225 = 4735 ; and
^•0022420225 = 04735, &c. : this is obvious. If there
is an odd number of decimal places in the power, it
must be made even by the add ition of a cypher. Using
the same figures, ^22420225= 1497338, &c.
224202256 (1497 338, &o
24)124
_%_
289)2820
2CM_
2987)2H)22
20909
29943)101350
89829
299463)1152100
898389 _
2994668)26371100
23957344
1413756
in this case the highest period consists but of a single digit
nilU flip frlVf>n linTYlVvisV lO »irif o ^n,<fnn4 <./>,,.'!«»
There must be an even number of decimal places ; .lince nc
number of decimals in the root will produce an odd numbe?
in thi^ square [Sec. II. 48]— as may be proved by experimen*
ia_Ji
312
EVOH
KXKR
JTION.
CISES.
20.
21.
22.
23.
24.
25.
14.
15.
10.
17.
yi95304=442
^328329— 573
^•0070= 26
^87 05=9 3022
^^801=29 3428
^984004=992
^5=2 23007
y 6= 707 100
V'Ol 9081— 959
.y 238 144=488
18.
10.
^^^2 3761=5 09
^•33 1770= 576
14. To extract the square root of a fraction —
EuLE. — Having reduced the fraction to its lowest
torni.s, make the square root of its numerator the nume
rator, and the square root of its donominatcr the deno
minator of the required root.
Example.— y*=f.
16. Reason of the Rule.— The square root of any quau
tity must bo such a number as. multiplied by itself, will pro
duce that quantity. Therefore f^ is the square root of  ; for
I y^ l=ff ^^e same might be shown by any other example.
Basides, to square a fraction, we must multiply its numera
tor by itself, and its denominator by itself [6] ; therefore, to
take its square root— that is, to bring back both numerator
and denominator to what they were before— we must take tbe
square root of each.
16. Or, when the numerator and denominator are
not squares —
Rule. — Multiply the numerator and denominator
together ; then make the square root of the product the
numerator of the require 1 root, and the given denomi
nator its denominator ; or make the square root of the
product the denominator of the requu'ed root, and the
given numerator its numerator.
Example.— What is the square root of f ? ()J a
=4472136{5='894427.
^/1X5 ^^
6 ^6X4
17. We, in this case, only multiply the numerator and
denominator by the same number, and then extract the square
root of each product. ^^^^ 5=5"^' or ^. Therefore ()^
''4x4 a 4
V5X5/
2
_s/^X5
— L, or
\5X4/
V5X4*
A>
EVOLUTION.
313
Ifi. Or, lastly—
lliTLE. — Hediico the given fraction to a decimal
[Seo IV. 63J, and extract its square root [13J
EXERCIHKS.
20 /22\i 285300852
27
28.
\37/ "^
37
14
14 '9000295
6244998
13/
13
29.
30.
Sli
(^)^=745350
(j^y.=:' 8000254
(f)'
8451542
19. To extract the square root of a mixed number —
Rule. — lleduce it to an improper fraction, and then
proceed as already directed [14, &c.]
Example.— y2.I=y^
=^=H.
EXERCISES.
32. y51j=71
33. y27VV=5i
34. yl ''o^lOlSSS
35. v'lI=llG83
3G. y_0,^=25298
37. ^'13^=30332
20. To find the cube root of any ni;..\ber —
Rule — I. Point oif the digits in threes, by dots —
putting the first dot over the units' place., and then
proceeding boi/i to the right and left hand, if there aro
digits at both sides of the decimal point.
II. Find the highest digit whose cube will not ex
ceed the highest period, or that which is to the left hanu
side — this will be the highest digit of the required root;
subtract its cube, and bring down the next period to
the remainder.
HE. Eind the highest digit, which, being multiplied
by 300 times the square of that part of the root,
already found — being squared and then multiplied by
30 times the part of the root already found — and being
multiplied by its own square — the su7)i of all the pro
ducts will not exceed the suvi of the last remainder and
the period brought down to it. — Put this digit in tho
root after what is already there, and subtract the former
ium from the latter.
IV. To what now remains, bring down the next
r ■'
314
EVOLUTION.
period, niul procooJ ns botorc. Continue tliia process
until tlio exact cube root, or a suflicieutly near ajtproxU
ination to it, is obtained.
ExAMi'LE.\Vhat i8 the cube root of 1795970G9288 ?
179597009288(5042, tho required root.
125
300x5»x0
30x5 xO»
G'XO
30()x50»x4
30x50x4»
4«x4
300x5G4'''x2
30x504x2^
2*x2
545!)
= 500
3981009
3790144
190925288
190925288
We find (by trial) tliat 5 is tho first, the second, 4 tho
third, and 2 tlio last digit of tho root. And the given
number is exactly a cube.
21. IIeason of I. — We point off the digits in threes, for a
reason similar to that which caused us to point thorn off in
tf?os, when extracting the square root [12].
Reason of II. — Each cube will be found to contain the
cube of each part of its cube root.
Reasoist of III. — The cube of a number divided into any
two parts, will be found to contaiu, besides the sum of the
cubes of its parts, tlie sum of 3 times the product of «ach
part by tlie otl.er part, and 3 times the product of each vart
by the squaio of tho other part. This will appear from the
following : —
179597069288
5000*=1 25000000000
54597069288
X 5000"' X GOOf 3 X 5000 X G00'}G00*= 5061 6000000
3 X 5000' X 40f 3 X 5G00 X 40^+40'
3981009288
3790144000
190925288
8 X 5640^ X 2 (8 X 5640 X 2*+2'= 1 90925288
Hence, to find the second digit of the root, we must find by
tnai some rrarnbcr which — being multiplied hj 3 times the
square of the part of the root already found — its square being
EVOLUTION.
315
mnltiplio'l l»y H Hmoa tlio part of tho root nlromly fotin.l— and
lii'itig iimltiplicd by tho nqunrc of UhoU'— tho Htim of the pro
ducts will not exceed wliat rornnins of tho j^fiven numhnr.
JiiHtoiuI of couHideriiif^ tlio part of tlio rnot iilretidy fdund ns
to many tens [i2J of the denoiuiiiatiou next fdllowiiig (jih it
rt'iilly Ih), which woidd (idd one cypher to it, ami two cyphers
to ItH aquaro, wo consider it as so many iinitH, and multiply
It, not )>y 3, but by HO, ami its Bquarc, not by %, but by 800.
For 800 X 5' X t5 i '"'^ X & X G'fG'X') Ih the sanio thing as
8xr>0'XGf3x50Xt»'+<)'X'»; since Ave only change tho posi
tion of the factora 100 and 10, which docs not alter tho product
[Sect. 11. 35].
It in evidently unncccHsnry to bring* down more than ono
period at a time ; or to add cypherB to tlie subtraliendH.
Ukasov ok IV. — The portion of tlie root already fniind may
be treated as if it "ro a sinfflo digit. 8inco into wliatever
two parts wo («livido any number, its cube root will contain
tlio cube of ench part, with o times the flquaro of each multi
plied into the other.
22. Whon there me. decimals in tho given cube —
If any of the periods consist of decimals, it is evident
that the difji;its found on bringinjjj down tbeso periods
Ljust be decimals. Thus ^17U.5'}7()6928S = 5n42, &c.
When the dtMJimals do not form complete periods, the
periods are to bo completed by the addition of cyphers.
ExABiPLE. — What is tho cube root of 3 '?
0'800(CG9, &c.
21G
800X6'X6
SOXGXG^
GXG'
800 X 66' X 9
80XGGX9'
0X9*
•669, &c. And
84000
=71496
12504000
=11922309
581G91, &c.
^•3='669, &c. And 3 is not exactly a cube.
It is ncce.ssary, in this case, to add cyphers; since ono decimal
in the root will give 3 decimal places in the cube; two decimal
^laces in the root will give six in the cube, &c. [Sec. II. 48.]
KXKRCISES.
88. yp=3 207534
89. 4/39=3 391211
40. y2r2=5962731
41 . ^n 23505'.!92=4 98
42. ^190r0U37"5=575
43. ;/458ai4011=771
44. ^ 483 • 736 (325=^7 85
45. ^•G3G05a=86
4(^ 3/099=') •ODGGGG
47. y 979140657= 993
i!
1
i
1
31G
EVOLUTION.
.2
23. To extract the cube root of a fraction —
3luLE. — JIaving reduced the giveu fraction to its
lowest terms, make tlie cube root of its numerator the
luimerator of the required fraction, and the cube root
of its denominator, the deuomiuator.
^''' ^125
21 Reason of the Rule.— The cube root of any number
must be such as that, taken three times as a factor, it will
procluce that number. Tlierelbre f is the cube root of  3^^;
fov j X I X f = yI ^.— Tlie same thing might be shown, by uuv
otiier example. ''
Resides, to cube a fraction, we must cube both numerator au<?
denominator; therefore, to take its cube root— tliat is to reduce
It to what it was before— wo must take the cube root of both.
25. Or, when the numerator and denominator are
not cubes —
llui.E. — IMuItiply the numerator by the square of tlu^
denominator ; and then divide the cube root of the pro
duct by the given denominator; or divide tlie given
numerator by the cube root of the product of the given
denominator multiplied by the square of the giveu
numerator.
Example.— What is the cube root of 5 ?
^— 2
or .^. = 5277032 ^ 7 == 753047.
(./ = ^3XP
5/7x3'
This vale depends on a principle already explained [IG].
26. Or, lastly—
Rule. — lloduce the given fraction to a decimal
[Sec. IV. 63], and extract its cube root [22] .
48.
40.
50.
8G5349(
\11/ ~5 604079
EXEnCISE.S.
61.
52.
7(>51725
(^y=560907
■472103
27. To fijid the cube root of a mixed number —
lluLE. — Iteduce it to :m improper fraction ; and then
proceed as already directed [i>3, &c.]
EVOLUTION.
317
EXERCISES.
54. (28ni=30G35
55. (7})J=l93098
56. (9^)i=20928
57. (71f)*=41553
58. (32/y)^=31987
59. (5)Ul7592
28. To extract any root wliatever —
liuLE. — When the index of the root is some power
of 2, extract the square root, when it is some power of 3,
extract the cube root o* the given number so many times,
Buccessively, as that power of 2, or 3 contains unity.
/
Example 1.— The 8th root of 65530=>/Vy 65536=4,
Since 8 is the third power of 2, we are to extract the
square root three times, successively.
Example 2.— 134217728«=yVl342lT7S=8.
Since 9 is the second power of 3, we are to extract the
cube root twice, suocessively.
29. In other cases we may use the following (Hutton
Mathemat. Diet. vol. i. p. 135).
Rule. — Find, by trial, some number which, raised
to the power indicated by the index of the given root,
will not be far from the given number. Then say,
as one less than the index of the root, multiplied by the
given number — plus one more than the index of the root,
multiplied by the assumed number raised to tlie power
expressed by the index of the root : one more than the
index of the root, multiplied by the given number —
plus one less than the index of the root, multiplied by
the assumed number raised to the power indicated by
the index of the root, : : the assumed root : a ^ still
nearer approximation. Treat the fourth proportional
thus obtained in the same way as the assumed number
was treated, and a still nearer approximation will be
found. Proceed thus until an approximation as near as
desirable is discovered.
Example.— Wliat is the 13th root of 923 1
Let 2 bo the assumed root, and the proportion will be
12x923+14x2^' : 14x923+12x2*^ :: 2 : a nearer
approximation. Substituting this nearer approximation for
2, in the above proportion, we get another approximation,
which wo may treat iu the same way.
318
EVOLUTION
EXKRCISK3.
GO, (9GG98)K=G7749
Gl. (GG457)iT=27l42
62. (23G5)?=31585
68. (8742G)?=508429
04. (89G5)'=l368
65. (•07542G)t4=04G988
30. To find the squares and cubes, the square and
cube roots of numbers, by means of the table at the end
of the treatise —
This table contains the squares and cubes, the square
Rnd cube roots of all numbers which do not exceed 1000
hut it will be found of considerable utility even when very
hi£':h numbers are concerned — provided the pupil bears
in^inind that [12] the square of ai\y number is equal to
the sum of the squares of its parts (which may be found
by the table) plus twice the product of each part by the
sura of all the others ; and that [2 1 ] the cube of a
number divided into any two parts is equal to the sum
of the cubes of its parts (which may be found by the
table) plus three times tne product of each part multi
plied by the square (found by means of the table) of
Hie other. One or two illustrations will render this
sufficiently clear.
Example 1. — Find the square of S734r)G.
873450 maybe divided into two parts, 873 (thousand) and
45G (units) . But we find by the table that 873'=7G2120 and
450'=20793G.
Therefore 762129000000=873000'
700176000=873000 X twice 45G
207936=450'
And 702025383936=873456'
ExAMPLK 2.— Find the cube of 864379. Dividint; this into
864 (thousand )_and 379 (units), wejfind 86?=(vi4972544
b'64 =746496, 379 =54439939, and 379 =143641
Therefore 644972544000000000=8(HOOO'
848765952000000=3 X 804 W X 379
3723 1 7472000=3 x 804000 x 3?j'
54439931
379
And G45821G82323911939=r86^
LOGARITHMS.
319
<5l In finding tlie square and cube roots of larger numbers,
we obtain their three highest digits at once, if we look in the
table for the Jiighest cube or square, the highest period of
which (the required cyphers being added) does not exceed the
hiohest period of the given number. The remainder of the
process, also, may often be greatly abbreviated by means of
ithe table. •
QUESTIONS.
1. What are involution and evolution } [1].
2. What are a power, index, and exponent > [1 & 2J.
3. What is the meaning of square and cube, of the
B(iuare and cube roots } [I and 8J.
4. What is the difference between an integral and a
fractional index .? [2 and 8] .
5. How is a number raised to any power } [5].
6. What is the rule for finding the square root } [10].
7. What is the rule for finding the cube root ? [20] .
8. How is the square or cube root of a fraction or
of a mixed number found > [14, &c., 19, 23, &c., 27].
9. How is any root found } [28 and 29] .
10. How are the squares and cubes, the square roots
and cube roots, of numbers found, by the table .? [30] .
LOGARTIHMS.
32. Logarithms are a set of artificial numbers, which
reprcsent°the ordinary or 'natural numbers. Taken
along with what is called the base of the system to
which they belong, they are the equals of the corres
ponding natural numbers, but without it, they are
merely their representatives. Since the base is un
changeable, it is not written along with tlie logarithm.
The logarithm of any number is that power of the base
which ts equni to it. Thus 10^ is eqital to 100 ; 10 is
the hase^ 2 (the index) is the logarithm^ and 100 is the
corresponding natural number.— Logarithms, therefore,
are merely the indices which designate certain powers
of some base.
33.. Logarithms afford peculiar facilities for calcu
lation. For, as we shall sec presently, the multiplica
tion of numbers is performed by the addition of their
320
LOGARITHMS.
logarithms ; one number is divided by another if we
subtract the logarithm of the divisor from that of the
dividend ; numbers are iuvolveu J' we multiply tJioir
logarithms by the index of the proposed power ; and
evolved if wo divide their logarithms by the index of
tho proposed root.— But it is evident that addition and
subtraction are much easier than multiplication and
division ; and that multiplication and division (particu
larly when the multipliers and divisors are very small)
are much easier than involution and evolution.
34. To use the properties of logarithms, they must bo
exponents of the same base— that is, the quantities raised
to those powers which they indicate must be the same.
Ihus 104X123 is neither 10^ nor 12% the former bein.
too small, the latter too great. If, therefore, we desirS
to multiply 104 and 12« by means of indim, we must
Imd .some power of 10 which will be equal to 123 or
some power of 12 which will be equal to 10% or finally
two powers of some other number which will be equal
respectively to 10^ and 123, ^^^ then, adding these
powers of the same number, we shall have that power
ot It which will represent the product of 10^ and 123
Ihis explains the necessity for a table of looarithms—
we are obliged to find the powers of some one base which
will be either equal to all possible numbers, or so neariy
equal that the inaccuracy is not deserving of notice The
base of the ordinary system is 19 ; but it is clear that
tiiere may be as many difierent systems of logarithms
as there are difierent bases, that is, as there are difierent
numbers.
35. In the ordinary system— which has been calcu
lated with great care, and with enormous labour, 1 is
the logarithm of 10 ; 2 that of 100 ; 3 that of 1000, &c
And, since to divide numbers by mejins of these loga
nthms (as wo shall find presently), we are to subtract
the logarithm of the divisor from that of the dividend,
IS the logarithm of 1, for 1=L^— 10''— 10" • — 1 Ls
10 '
the logarithm of 1, for •l=si=lo''=10«— 1^=^10
10 101
for the same reason, 2 is the logarithm of 01 :
that ol 001, &c. '
1 . „„.!
, uuu
bcr, a
LOOAUITHMS.
521
^■', or
'3
36. The logaiitlinis of numbers hetwem 1
must 'be more tlifin and less than 1 ; that is,
Bomo decimal.
and 1 0,
, Kn.iu xn, must bo
The logarithms of numbers between JO
and 100 must be more than 1, and less than 2 ; that
is, unity with some decimal, &c. ; and the logarithms of
numbers between 1 and Ql must be —1 and"' some deci
mal ; between 01 and 001, —2 and some decimal, &c.
The decimal part of a logarithm is ahoays positive.
37. As the integral part or charaderisik of a posi
tive logarithm is so easily found — being [35] one less
than tlie number of integers in its corresponding num*
bcr, and of a negative logarithm one more than thu
number of cyphers prefixed in its natural nuniber,
it is not set down in the tables. Thus the logarithm
corresponding to the digits 9872 (that is, its decimal
part) is 99440:'^ ; hence, the logarithm of 9872 is 3
•994405 ; that of 9S72 is 2994405 ; that of 9872 is
0994405 ; that of 9872 is 1994405 (since there is no
integer, nor prefixed cypher) ; of 009872— 3*994405,
&c. : — The same digits, whatever may be their value,
have i\\Q EAmQ decimals in their logarithms; since it
is the integral part, only, which changes. Thus the
logarithm of 57864000 is 7702408 ; that of 57864, is
47G2408 ; and that of 0000057864, is— 6762408.
38. To find the logarithm of a given number, by the
table —
Tlie integral part, or charaGtoristic, of the logarithm
may be found at once, fioni v/liat has been just said [37] —
When the number is not greater than 100, it will bo
found in tlie column at the top of which is N, and the
decimal part of its logarithm iinmediately opposite to it
in the next column to tlio light liand.
If the number is greater than 100, and less than
1000, it will also bo found in the column marked N,
and the decimal part of its logarithm opposite to it, iu
the column at tlie top of which is 0.
If the number contains 4 digits, the first three of
them will be found in the column under N, and th«
fourth at the top of the pngo ; and tlion its logarithm
in Jic same horizontal lino as the thi(!o first digits of
the given nunibor, and in the same column as its fourth
ill
v
322
LOGARITHMS.
^
If the number contains more than 4 digits, find the
logarilhui of its first, four, and ;ilso the diifcrence be
tween that and the h)gaiithni of tiic next higher num
ber, in the table ; multiply this diiFereuue by the remain
ing digits, and cutting off from the pr(^uct so many
digits as were in the multiplier (but at the same time
ftdding unity if the highest cut off is not less than 5), add
it to thcr logarithm corresponding to the four first digits.
Example 1.— The logarithm of 59 is !• 770852 (the charac
teristic being positive, and 07w less than the number oiintegers) .
Example 2.— The logarithm of 338 is 2528917.
Example 3.— The logarithm of 0004587 is — 4(561529
(tlie characteristic being negative, and one moix than the
number of prefixed cyphers) .
Example 4.— The logarithm of 28434 is 4453838.
For, the difference between 453777 the logarithm of 2843,
the four first digits of the given number, and 453930 the
logarithm of 2844, the next number, is 153 ; which, multi
plied by 4, the remaining digit of the given number, pro
duces G12: then cutting off one digit from this (since we
have multiplied by only one digit) it becomes Gl, which being
added to 453777 (the logaritlim of 2844) makes 453838, and,
with the characteristic, 4453838, the required logarithm.
Example 5.— The logarithm of 873457 is 5941242.
For, the difference between the logarithms of 8734 and
8735 is 50, which, being multiplied by 57, the remaining
digits of the given number, makes 2850; from this we cut
off two digits to the right (since we have multiplied by two
digits), when it becomes 28 ; but as the highest digit cut
off is 5, we add unity, which makes 29. Then 5941213 (the
logarithm of 8734) [29=5941242, is the required logarithm.
39. Except when the logarithms increase very ra
pidly — that is, at the commencement of the table — the
differences may be taken from the right hand column
(and opposite the three first digits of the given number)
where the mean differences will be found.
Instead of multiplying the mean difference by the
remaining digits (the fifth, &c., to the right) of the given
number, and cutting off so many places frc^ the product
as are equal to the number of digits in the multiplier,
tx) obtain the iir)pur!luaal part — or what is to be added
5, 2u.
LOGARITHMS.
323
to the logaritlim of tlie first four digits, we may tako
the ^oportioiuil part corrcspouding to each of the re
inaitiing digits from that part of the columu at the left
hand side of the page, which is in tlio same horizontal
division as tliat in which the first three digits of the
givvon number have been found.
K.VAMi'Li:.— What is the logarithm of 839785 ?
The (decimal part of the) logarithm of 8B9700 is 924124.
Opposite to 8, in the same horizontal division of the page,
wo lind 42, or rather, (since it ia 80) 420, and opposite to
5, 2u. Monce the re(iuirod logarithm'is 9241244420f2G=«:
Vi24570: uud, with the characteristic, 5924570.
40. Tlic mctliod given for finding tlie proportional part — or
what is to bo added to the next lower logarithm, in the table—
iirisos from tlie diiferoiico of numbers being proportional to the
ditference of their logarithms. Hence, using the last example,
100 : 86 : : 62 (92417*i, the logarithm of 839800—924124,
the logarithm of 839700) : ""Vqq. or the difference (the ynean
difference mnj generally be used) X by the remaming digits of
tlie given nuiMl)or — 100 (the division being performed by cut
ting off two digits to the riglit). It is evident that the number
of (iigits to be cut off depends on the nuniber of digits in the
multiplier. The logaritlim found is not exactly correct, be
cause numbers are not exactly proportional to the difforcncca
of their logarithms.
The proportional parts set down in the left hand column,
have been calculated by making the necessary multiplica
tions and divisions.
41. To find the logaritjiini of a fraction —
lluLE. — Find the logarithms of both numerator and
denominator, and tlien subtract the former from tho
latter ; this will give the logarithm of the quotient.
Example.— Log.' i is 1672098  1748187 =  1923910.
Wo find that 2 is to be subtracted from 1 (the character
istic of the numerator) ; l)nt 2 from 1 leaves 1 still to bo
snbtractcd, or [Sect. II. 15 j — 1, the characteristic of tho
quotient.
Wo shall find presently tliat to divide one quantity bj
anoth.or, avo have merely to subtract the lop;arithm of the I'attei
iVoni that of tho former.
42. To find the logarithm of a mixed number —
iluT.K. — llcduee it to an improper fraction, and pro
c<"od as directed by the last rule.
i
/'■
324
LOOARITIIMS.
43. To fiud the numjicr which corresponds to a given
logarithm — • '"•
If the logarithm itself is found in the tahle —
lluLE. — Take from the table the number which cor
responds to it, and place the decimal point so that there
may be the requisite number of integral, or decimal
places — according to the characteristic [37].
Fa' AMPLE. — What number corresponds to the logarithm
4214314?
AVo find 21 opposite the natural number 103 ; and look
i g along the horizontal line, we find the rest of the logarithm
under the figure 8 nt the top of the page : therefore the digits
of the required number are 1038. But as the charaeteribtic
is 4. there must in it be 5 places of integers. Hence the
required number is 1G380.
44. ,If the given logarithm is not found in the table — ■
liuLE. — Find that logarithm in the table which is
next lower than the given one, and its digits will bo
the highest digits of the required number ; find tho
diflerence between this logarithm and the given one,
annex to it a ryphcr, and then divide it by that differ
ence in the table, which corresponds to the four highest
digits of the required number — the quotient will be the
next digit ; add another cypher, divide again by the
tabular diflerence, and the quotient will be th: next
digit. Continue this process as long as necessary.
]'2xAi\rPLE. — What number corresponds to the logarithm
5054329 1
C54273, which corresponds with the natural number 4511,
is the logarithm next less than the given one ; therefore the
first four digits of the required number are 4511. Adding
a cypher to 50, the difference between 054273 and the given
logarithm, it becomes 500, which, being divided by 90, tlie
kihidar difference corresponding with 4511, gives 5 as quo
tient, and 80 as remainder, I'herefore, the first five digits
of the required number are 45115. Adding a cyplier to 80,
it becomes 800; and, dividing this by 90, we obtain 8 as
the next digit of the required number, and 32 as remainder,
"^riio iiife'rcrs of the required numl^or (one more than 5, tho
characteristic) are, tlierofore, 451158. We may obtain the
decimals, by continuing the addition of cyphers to the re
mainders, and the division by 90.
4o.
V
X.
LOQARITIIMa.
325
45. Wc arrive at tlio same lesult, by Bubtracting
from .the difference between the given logarithm and
the next less in the table, the highest (which doesno
exceed it) of those proportional parts found at the right
W side^f the page and in the same honzontaWm
sion with the first three digits of the given number
continuing the process by the addition of cyphers, until
nothing, or almost nothing, remains.
FxAMPLK.Usin the last, 4511 is the natural number
cor esTonUng to the logarithm G54273, which differs from
he ^enlolarithmby'sG. The Pvoptf.r^^^^^ %'
Bimo horizontal division as 4511, are 10, 19, 2J, c5», 4», oo,
G7 77 md 80. The highest of those, contained in 56, is
48. w I'ich we find opposit^e to, and therefore corresponding
with tho natural number 5; hence 5 is the next of the
•0 ulred digits. 48 subtracted from 5G, leaves 8 ; this, when
a'^^pher is^dded, becomes 80, which contains 77 ^corres
w>udino to the natural number 8)5 therefore 8 is the next
^ the "required digits. 77 subtracted fm ^O, kaves 3
tliiK when a cypher is added, becomes 30, &c. ^^^o inte
to 5 Therefore, of the required number, are found to be
451158, the sauie as those obtained by the other method.
The rules for finding the numbers corresponding to
civon looarithms are merely the converse of those used
for finding the logarithms of given numbers.
Use of Logarithms in Arithmtic.
46. To multiply numbers, by means of their loga
" KmlAdd the logarithms of the factors ; and the
natural number corresponding to the result will be the
required product.
ExAMPLF..87x24=1939519 (the log. of 87) f 1380211
Ohe^olof 24)=3319730; which i^fo^^^T ° ^rXx
ivith the natural number, 2088. Therefore 87x24=2(588.
from the very nature of indices. Thus f X° — ^0'^,?^'^O^'^
jrom uie vLi^y and the abbreviation for
multii.liodSXoXoXoXoXoxoxJ, I indices
(logatulnlv "The rule rnighl in the same way, be proved
correct by any other example.
p Q
I
326
L0CiAlUTJIM8
47. When tho clmractcristies of tlio logarithms^ to be
added arc both i)ositivo, it is cvidont that their sum will
bo positive. When thoy are both negative, their sum
(diminished by wliat is to bo carried from tlie sum of
th.i positive [36] decimal parts) will be negative. When
one is negative, and tho other positive, subtract tho less
from the greater, and prefix to the difference the Bign
belonging to tho greater — bearing in mind what has
been already said [Sec. II. 15] with reference to the
subtraction of a greater from a less quantity.
48. To divide numbers, by means of their logarithms —
liui.R. — Subtract tho logarithm of the divisor from
that of the dividend ; and tho natural number, corres
ponding to tho result, will be tho required quotient.
Example.— 1134 f. 42=3054013 (the 1o;t. of 1184) —
1G23249 (the log. of 42) = 1431304, which is found to
correspond with the natural number, 27. Therefore 1134—
42=27.
Reasox of the Rule. — This mode of division arises from
the nature of indices. Thus 4*i4'=[2] 4x4X4X4X4— IX
4X4X4X4X4 ^ ^ 4x4x4 ^
4X4= 4>^4x4 — =4X4X4^^1=4x4, the abbreviation
for which is 4»; Bu*, 2 is equal to tho index (logarithm) of
the dividend minus hat of the divisor. Tlie rule might, in
the same way, be provvd correct by any other example.
49. In subtracting tho logarithm of the divisor, if it
is negative, change the sign of its characteristic or inte
gral part, and then proceed as if this were to be added
to the characteristic of tho dividend ; but before making
the characteristic of the divisor positive, subtract what
was borrowed^ (if any thing), in subtracting its decimal
part.^ For, since the decimal part of a logarithm is
positive, what is borrowed^ in order to make it possible
to subtvact the decimal part of the logarithm of the
divisor from that of the dividend, must be so much
taken away from what is positive, or added to what '3
negative in the remainder.
We chac^ge the sipin of tho negative characteristic, ana
then add it; for, adding a positive, is the same as taking
awny a negative quantity.
V
Ill
i,o«AurrHM«.
fsrr
in
^0. To also a quantity to any power, by means of
it» logaritli.n —
lluLE. — Multiply tho lofrarltlim of tlie qiianity by
tho index Oi t\w power; and the natural number cor
rcbponding to the re.sult will be the required power.
KxAMPi.E. — Puufic 5 to tho 5th power.
The lo^'arithm of 5 is OO'.IHOT, whieh, niultlpllcd by 5,
gives 3'4'J486, the logarithm of 3125. Theroforo, the 5th
jiowcr of 5' is 3125.
kr.AHON OK Tiiii Rui.K. — Tliis vnlc also follows from tlie
nawire of imlicea. o* vnisod to tlie otli power is 6X& inuUi])li»>d
hy Hx'o )uulti{)lieil by oX5 nuillii;licil by SX) nmltiplieil
by rjX5, or r)X'''X5xr)X''iX'''>Xi)X'^X':>X0, the abbreviation
for which is [2] 5'". lUit 10 is eiiiial to 2, the index (b.garithm)
of tho quantity, niultiplioa by 5, that of the ])Ower. The
rnle might, in the same way, bo proved correct by uuy other
example.
51. It follows from what has been said [47] tbitt wbcu
a negative chariieteristic is to be multiplied, the produet
in nrrrativc ; and that what i:< to be carried from the
nmltijljcation of the decimal part (always positive) is
to be suhtraclcd from tlii.s mj^ativo result.
52. To evolve any quantity, by means of its loga
rithm —
Bulk. — Divide the logarithm of the given quantity
by that number which expresses the root to be taken ;
and the natural number corresponding to the result will
be the re(uired root.
Ex.ypi.E.— What is the 4th root of 2401.
Tlie lo;i;arithm of 2401 is 3 o80302, which, divided by 4,
the number expressing the root, gives 845098, the logarithm
of 7. Therefore, the fourth root of 2401 is 7,
Heason ok the Rule. — This vule follows, likewise, from
tho nature of indices. Thus the 5th root of ItV ia such a
number as, raised to the 5th power— tliat is, taken 5 times aa
<i factor— would produce 16'". But loV, taken 5 times as a
factor, would produce 10'". The rnle might be prove 1 correct,
equally well, by any other example.
53. "When a negative characteristic is to bo divided—
UiTi.i: I.— If the cliaractevistic is cKadly divi.sible by
the divisor, divide in the ordinary way, but make tho
characteristic of the quotient negative.
I
III
1
A
1^4
m
Wm
328
LOGARITHMS.
TT. — Tf llio nogativo clmractcristio is nni exactly
rlivisiblc, juJd wluit will iimko it so, both to it and to the
decimal part of tho logarithiu. Theu proceed with tlu
division.
Example.— Divide tho logarithm —4' 887564 by 5.
4 w antH 1 of bciug divi«iblo by 5; then — 4•8375C4^5=a
— 5f.l8375G4i5=13G7513, tho required logarithm. *
Rraaon of I. — Tho quotient multiplied by tho divisor must
give tlio dividend; but [61] a ncgativo quotient multiplied by
a positive divisor will give a negative dividend.
Kkahon of II. — In cxanii>le 2, avc luivo merely added f 1
and — 1 to tho same quantity — wliich, of course, docs nC
.alter it.
QUESTIONS.
1 . What are logarithms ? [32", .
2. How do they facilitate calculation .? [33] .
3. Why ia a table of logarithms necessary } [34].
4. What is the characteristic of a logarithm ; ant/
how is it found ? [37] .
5. IIoAv ih the logarithm of a number found#by tho
table.? [38].
f). How are the " diflferenccs," given in the tablo
used.? [30].
7. What is the use of "proportional parts .?" [39].
S. How is the logarithm of a fraction found ? [41].
9. How do we find the logaritlim of a mixed num
ber ? [42] .
10. How is tho number corresponding to a given
logarithm found ? [43] .
11. How is a number found when its corresponding
logarithm is not in the table ? [44].
J2. How are multiplication, division, involution and
evolution effected, by means of logarithms r [46, 48,
50, and 52].
13. When negative characteristics are added, what
is tlie sign of their sum ? [47].
14. What is tlio process for division, when tho cha
racieristic of the divisor is negative ? [49] .
15. How is ancgTitivc characteristic umltiplied r [51].
10. How is a negative charactori.stio divided .? [53]
5
5
329
SECTION X.
PROGRESSION, &o.
1. A profTreasion consi.sts of <*i nnmbnr of quantities
Jill cm a sing, or decreasing l)y fi ciirtaiu law, and forming
^vliat uro called cmitmued propor/Umnts. When the
terms of the series coui*tantly increase, it is said to
l>«) an ascending^ but when they decrease (increase to
(he /rft), a descending scries.
2. in an fqnidiO'crcnt or a ?i/Awt'/(!"mZ progression, tho
qnantities inciease, or decrease by a annvion difference.
Tiius 5, 7, 9, 11, &.O., is an ascending, and 15, 12, 9, 6,
&c., ia a descending arithmdical series or progrtission.
The common diflerencc in the former is 2, and ni the
latter '^. A continued proportion may bo formed out
T)f such a series. Thus —
5 : 7 : : 7 : 9 : : 9 : 11, &c. ; and 15 : 12 : : 12 : 9 : :
9 : 0, &o. Or we may say 5 : 7 : : 9 : 11 : : &c. ) and
15 : 12 :: 9 : 6 :: &c.
3. In a geometrical or equirallonal progression, tho
quantities increase by a common ratio or multiplier.
Thus 5, 10, 20, 40, &c. ; and 10000, 1000, 100, 10, &c.,
are geometrical series. The common ratio in the former
case is 2, and the quantities increase to the right ; in
the latter it is 10, and tho quantities increase to tho
left. A continued proportion may be formed out of
Huch a series. Thus —
5 : 10 : : 10 : 20 : : 20 : 40, &c. ; and 10000 : 1000 : :
1000 : 100 : : 100 : 10, &c. Or wo may say 5 : 10 : : 20 :
40 : : &c. ; and 10000 : 1000 : : 100 : 10 : : &o.
4. The first and last terras of a progression are called
its extremes, and all the intermediate terms its means.
5. AritliMeiiad rrogression.—To find the sum of a
series of terms in arithmetical progression —
j^^yij,;, — Multiply the sum of the extremes uj nan
the number of terms.
I
330
PROGRESSION.
_ Example.— Whafe is the sum of a series of 10 terms, tlio
tirat being 2, and last 20 '? Ans. 2f 20x •2'=110.
C. Reasoiv of the RuLic.— This rule can be easily proved.
For tins purpose, set down the progression twice over— but
in sucli a way as that the last term of one shall be under the
nrst term of the other series.
Then, 24+21+18415412f 9=the sum.
9f 12f15 418i21+24=the aum. And,
adding the equals, 3;J+33j33433+33+33=twice the sum.
That is, tf^ir.e tlie sum of the series will be equal to the sum
of aa ii.any quantities as tliere are terms in the series— each
of the quantities being equal to tlie sura of the extremes.
And the sum of the series itself will be equal to half as much,
or to the sum of the extremes taken ha/f as many times as
tliere are terms in flie series. The rule might be proved
correct by any other example, and, therefore, is general.
EXERCISES.
1. One extreme is 3, the other 1.5, and the number
of terius is 7. What is the sum of the series ? Ans. 63.
2. One extreme is 5, the other 93, and the number
of terms is 41). ^VHiat is the sum .? Ans. 2401.
3. One extreme is 147, the other , and the number*
of terms is 97. What is tlie sum ? Ans. 7165875.
4. One extreme is 4^, the other 143, and the num
ber of terms is 42. What is the sum > Ans. 3094875
7. Given the extremes, and number of terms — to fini
the common difference —
lluLE. — Find the difference between the given ex
tremes, and divide it by one less than the number of
terms. The quotient will bo the common difference.
Example.— In an arithmetical series, the extremes are 21
and o, and the nunibcr of terms is 7. Wha^; is the common
diilerence '?
21 — 3^7 — 1 = 18i6 = 3, the required number.
8. Reasov of thk Rule.— The diflference between the
greater and lesser extretne arises from the common ditferenco.
bemg aiJded to the lesser extreme once for every term, ex
cept tbe lowest ; that is, the greater contains the lesser extreme
plus the common difference taken once less than the number
of terms. Therefore, if we subrract the lesser from the greater
extrcriiu, the diifereuce oblaiuod will be equal to the common
dil}vMen'»e multiplied by one less than the number of terms
And if wo divide tlio difference by one less than the number
of tcrm.^ we will have the cuininun difference.
5.
Tb
and 497,
common
6. Th
and 9, :
common
7. Th
and I, a:
common
9. To
two give
Rule
cording
it to, or
term ; a
the thirc
ing tern
Wen
terms is
EXAMI
21. 21
the seri(
6 .
Exam
10. 30
the Borl
8. r
A:n3. 4
9. 1
Ans. 1
10
Am. 6
PnOftRESSION.
331
EXERCISES.
5. The extremes of an arithmeti(;al series are 21
and 497, and the number of terras is 41. What is the
common difference .? Aiis. 119. ^ mro,
6 The extremes of an arithmetical serieR are 127«a
and 91, and the. number of terms is 26. What is the
common difrerenee ? Aiis. 4^. ^
7 The extremes of an arithmetical series are 77f
and*!, and the number of terms is 84. What is the
common difference ? A^is. f .
9. To find a7ii/ number of arithmetical means between
two given numbers —
Rule.— Find the common difference [7] ; and, ac
cordinc' as it is an ascending or a descending series, add
it to, or subtract it from the first, to form the f oond
term ; add it to, or subtract it from the second, to torni
the third. Proceed in the same way with the remain
ing terms. , , i. <?
VVe must remember that one less than the number ot
terms is one more than the number of means.
Example l.~Find 4 arithmetical means between 6 and
21. 216 = 15. TTT==''^) th^' common difference. And
the series is — „ « . r o
6.6+3. 6+2x3 . 6+3x3 . 6+4x3 . 6+5x3.
Or 6 . 9 . 12 . 15 . 18 . 21.
Example 2.— Find 4 arithmetical means between 30 and
10 3010=20. TITT^^' *^^ common difference. And
4+i
the Bcrics ia — io ^ i i in
30 . 26 . 22 . 18 . 14 . 10
This rule is eyideut.
EXERCISES.
8 Find 11 arithmetical means between 2 and 26
A:m. 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24.
9 Find 7 arithmetical means Detween 8 and 64
Ans. 11, 14, 17,20, 23,26, 29.
10 Find b arithractieal means between 4J, and id^
Am. 6,7J,9, 101, 12.
332
rUOGRESSION.
10. Given the extremes, and tlie number of terms —
to find any term of an arithmetical progression —
Rule. — Find the common difference by the last rule,
and if it is an ascending series, the required term will
be the lessor extreme plus — if a descending series, the
greater extreme minus the common difference multiplied
by one less than the number of the term.
Example 1. — What is the 5th term of a series containing
9 terms, the first being 4, and the last 28 ?
284 ^
— Q — =3, is the common difference. And 4f 3x5 — 1=
16, is the required term.
Example 2. — What is the 7th term of a series of 10 terms,
the extremes being 20 and 2 ?
202 ^ _ „
— n— =2, is the common difference. 20 — 2x7—1=8.
is the required term.
11. Reason OF the Rxtle. — In an ascending series th»
required term is greater than the given lesser extreme to the
amount of all the differences found in it. But the number of
differences it contains is equal only to the number of lerir.a
which precede it — since the common difference is not found in
the^trs^ term.
In a descending series the required term is less than the
given greater extreme, to the amount of the differences sub
tracted from the greater extreme — but one has been subtracted
from it, for each of the terms which precede the required term.
exerclses.
11. In an arithmetical progression the extremes are
14 and 86. and the number of terms is 19. What is
the 11th term .? Ans. 54.
12. In an arithmetical series the extremes are 23 and
4j and the number of terms is 7. What is the 4th
ttrm } Ans. 13.
13. In an arithmetical series 49 and £ are the ex
tremes, and 106 is the number of terms. What is the
94th term } A^is. 62643.
12. Given the extremes, and common difference— to
find the number of terms —
11 ule.— 'Divide i\\Q differcnco bctAvccn the given ex
tremes by the common difference, and the quotient plus
unity will be the number of terms.
Ex A Mr 1
which th
ence 3 '?
20
3
13. He,
lesser ext
terms, i
except th
the extre
■will be C3
14. Ii
and 12,
number
15. Ii
and 32,
number
16. I
13
'<.}•)
an I
number
14. G
and one
llULE
terms, n
The difi
EXAMI
the nun
AVhat is
15. Rj
Slim = 8
divide ec
vre shall
2 X
the num
tractiug
have
rKOGREHSION.
833
ExAMiM.K. — How many tenns in jin firUhinotical series of
which tho extromea arc 5 and 20, and Iho common ditfer
eiicc 3 ?
205
3
=7. And 741=8, is tlie number of terms.
13. llEAscN OF THE Rui.E.— The srcatcr diifcrs from tho
lesser extreme to tlie amount of the diircrcncos found in all the
terms. But th» common difference is found in all the terms
except the lesser extreme. Therefore the difference between
the extremes contains the common difference once less than
will be expressed by the number of terms.
EXERCISES.
14. In an arithmetical series, tlie extremes arc 96
and 12, and tlie common difference is 6. What is tha
number of terms ? Ans. 15.
15. In an arithmetical series, the extremes are 14
and 32, and the common difference is 3. What is the
number of terms } Ans. 7.
16. In an arithmetical series, the common difference
is A, and the extremes are 14f and il. What is the
number of terms ? Ans. 8.
14. Given the sum of the series, the number of terms,
and one extreme — to find the other —
lIui.E. — Divide twice the sum by the number of
terms, and take the given extreme from the quotient
The difference will be the required extreme.
Example.— One extreme of an arithmetical scries is 10
the number of terms is 6, and the sum of the series is 42
What is the other extreme 1
2X42
10 = 4, is the required extreme.
15. Reason of the Rule.— We have seen [5] that 2 X the
sum = sum of the extremes X the number of terms. But if wo
divide each of these equal quantities by the number of terms,
■we shall have , o.
2 X the sum sura of extremes X the number of terms
the number of terras
2 X the sum
the number of terms
, — „ =s sum of the extreme!?. .And sub
the number oi terms
tractiug the same extreme from each of these equals, wo shall
have
ur
H34
i>K()(;kxs«u)!v.
„ X ''"^J^"J^ — onocxtioaie=the sum of the extremes
the number of tonus
tlie aaiue extreme.
twice the sum
Or ;^i.: „i rrr.:F7;~:z minus one extreme = the other ox
tho number of terms
aerae.
EXERCISES.
17. One extreme is 4, the number of terms is 17,
/md the sum of the series is 884. What is the other
extrcuio ? Ans. 100.
18. One extreme is 3, the number of terms is 63,
and the sum of the series is 252. What is the other
extreme ? A71S. 5.
19. One extreme is 27, the number of terms is 26,
and tlie sum of the scries is 1924. What is the other
extreme.? Ans. 121.
16.^ Geometrical Progression. — Given the extremes
and common ratio — to find the sum of the series —
Rule. — Subtract the lesser extreme from the product
of the greater and the common ratio ; and divide the
difference by one kss than the common ratio.
Example. — In a geometrical progression, 4 and 312 are
the extremes, and 'he common ratio is 2. What is the sum
of the series.
312x2 — 4
— 2 — Y — ~ ^= ^^^' *^® required number.
17. Reason of the Rui^e. — The rule may be proved by
setting down the series, and placing over it (but in a reverse
order) the product of each of the terms and the common ratio.
Then
Sum X common ratfo = 8 4 16 + 32, &c. . + 81 2 f 624
Sum= 4 + 8 + 16+32, &c. . 4312 .
And, subtracting the lower from the upper lino, we shall have
Sum X common ratio — Sum = 624 — 4. Or
Common ratio — 1 X Sum = 624 — 4.
And, dividing each of the equal quantities by the common
ratio minus 1
642 (last term X comrrfon ratio") — 4 (the firgt term)
Sun = ^^ ^ vv— — 5, — ^^ ~
common ratio — 1
Which ib the rule.
20. The
2, and th
Ans. 682.
21. The
175092, an
Ans. 1932
22. The
ore yV anc
the sum.?
Since the
23. The
9375, and
A.)is. 1171
18. Giv
goometricii
RULE.
by thc! les!
i.s indicate
bo the req
EXAMPIF
pro<:;rcs8U)v
11 1011 ratio 'i
SO
19. Tiv.^u
to the Icsse
tlie comuioi'
pinuo the c>
is, tlie gren
power imlic
tiplio'l by t
by the I0S.S1
is indicate;
obtain the
24. Th
common 1
85. Til
I'R0(iRE.S3I0N
335
EXERCISKS.
20. The extremes of a geometrical pciies are 512 and
2, and the common ratio is 4. What is the sum ?
Ans. 682.
21. The extremes of a geometrical series are 12 and
175092, and the common ratio is 1 1. What Is the sum ?
Ana. 193200.
22. The extremes of an infinite geometrical series
ore yV and 0, and j\ is tlie common ratio. What is
the sum.? Ans. i. [Sec. IV. 74.]
yinco the sorics is infinite, the lessor extreme=0.
23. The extren\es of a geometrical series are *3 and
9375, and the coiumou ratio is 5. What is the sum ?
A)is. 1171875.
18. Given the extremes, and number of terms in a
geometrical series — to find the common ratio —
EuLE. — Divide the greater of the given extremes
by the lesser ; and take that root of the quotient which
i.s indicated by the number of terms minuni 1. This will
be the required number.
ExAMPiE.— 5 and 80 arc the extremes of a geometrieal
progression, in which there arc 5 terms. ^Vhat i^ the eoiu
11 ion ratio ?
7=lG. And ^'1G=2, the required common ratio.
o
19. REASON OT THK Rule. — The greater extvemc is o.innl
to the lesser multiplied by a product whicli has for its factors
tlie conmioii ratio tukeri once loss tiian the number of terms —
since the comni<in ratio is not found in tiie Jiist term. That
is, tlie trreater cxtre\ae contains the common ratio raised to a
power indicated by 1 less than the number of terms, and mul
tiplied by the lessor extreme. Consequently if, after dividinj;
by the lessor extreme, we take that root of the quotient, which
is indicated by one less tlian the number of terms, we sholl
obtain the common ratio itself.
EXERCISES
24. The extremes of a geometrical series are 4911^2
oj^fl 3^ atul the number of terms is 8. What is th3
common ratio } Ans. 4.
2d. The extreuies of a geometrical series arc 1 and
336
PROGRKSSION,
15625, and the number of terms is 7 What is the
common ratio ? Ans. 5.
26. The extremes of a geometrical series arc
20176S035 and 5, and the number of term* is 10
What is the common ratio ? Ans. 7.
20. To find any number of geometrical means be
two on two quantities —
Rule. — Find the common ratio (by the last rule)^
and — accovdina; as the series is ascendinoj, or descend
ing — multiply or divide it into the first term to obtain
the second ; multiply or divide it into the second ta
obtain the third ; and so on with the remaining terms.
We must remember that one less than the number
of terms is one more than the number of means.
Example 1. — Find 3 geometrical means between 1 and
81.,
^=:3, the common ratio. And 3, 9, 27, are the re
quired means.
EXAiMPLE 2. — Find 3 geometrical means between 12r)f
und 2.
1250 _ . . 1250 1250 1250
4/2=''> And
H, or 250, 50, ]';'
5 5x5 5x5x5
are the required means.
This rule requires no explanation.
EXERCISES.
27. Find 7 geometrical means between Sand 19683 '
Ans. 9, 27, 81, 243, 729, 2187, 6561.
28. Find S geometrical means between 4096 and Si"
Ans. 2048, 1024, 512, 256, 128, 64, 32, and 16.
29. Find 7 geometrical means between 14 and
23514624.? Ans. 84, 504, 3024,
18144, 108864,
653184, and 3919104.
21 . Given the first and last term, and the number of
terms — to find any term of a geometrical series —
lluLE. — If it bo an ascendhig series, multiply, if a
df'sp.or>diTi'» sorins). divJ;<ft tho firs.t, tomi hv> fl 'ij imwrrr
of the common ratio which is indicated by t;*; num&ei
of the term minus 1.
is the
30.
32.
PROGKESSION.
337
Example 1.— Find tlio 3nl term of a {reometrical scries,
of %vhich the tiryt term is (3, the last 1458, suid the number
ol" term.) G.
1458
The common ratio is ;j/^=P). Tlicveforc the required
term is 0x3^=54.
]'>xAMPi,K 2. — Find the 5th term of a series, of which tlw
extremes are 524288 and 2, and the nundjer of terms is 10.
5242SH . 5242NS
Iho common ratio ^ — rr~— 4. And — j4 = 2048
is the required term.
22. Rkasoiv of the Riti.k.— Tn an ascending series, any
term is the proihict of the lirsi and the couinion ratio taken
as a factor so many times as there arc preceding terms — siiKsa
it is not found i7i ihe fust term.
In adeccndiug series, nny term is eqnal to the first term,
divided by a product containing the common ratio as a factor
so many times as tliero arc pieeoding terms — since eveiy term
but that whieh is required adds it once to the factors wducli
coDstitute the divisor.
EXERCISES.
30. Wh;it is tho 6th term of a series having 3 and
5851)375 as extremes, and, containing 10 terms .^ Ans.
9375.
31. Given 39366 and 2 as tho extremes of a series
Laving 10 terms. What is the 8th term.? Ans. 18.
32. Given 1959552 and 7 as tlio extremes of a series
havinff 8 terms. What is the Gth term ? Ans. 252.
O
23. Given the extremes and common ratio — to find
the number of terms —
Rule. — Divide the greater by the lesser extreme,
and one more than the number expressing what power
of common ratio is equal to the quotient, will be tho
re(ulrcd qiumtity.
ExAMi'Lic. — How many terms in a series of which tho
extriMnes are 2 and 25G, and the conuuou ratio is 2 ?
,^=128. But 2"=::128. There are, therefore, 8 term£3.
The common ratio is fonnd as a factor (in the quotient of
tho greater divided by tho lo.'^ser extreme) once less than the
number of terms.
II L
838
PROGRESSION.
KXERCISES.
33. TIow many terms in a xories of wliich the first, is
78732 and the last 12, and the cummun ratio is 9 .*
Ann. 5.
34. IIoAV niany terms in a series of wliieli llic ex
tremes and common ratio are 4, 47()r)9f;, and 7 ? Avs. 7.
35. How inany terms in a series of Avliieh the ex
tremes and common ratio, are 19GGUb!, G, and 8 f Ans. 6.
24. Oiv^en the common ratio, number of terms, and
one extreme — to find the other —
lluLE. — If the lesser extreme is given, multiply, if
the greater, divide it by the common ratio raised to a
power indicated by one less than the number of terms.
ExajMpi.k 1. — In a g;oomctrioal series, the lesser extreme
is 8, the number of terms is 5. and the enniinoii ratio is Gj
what is the other extreme '? Atis. 8xG'~'=10oG8.
Example 2. — In a goomefrioal series, the greater extreme
is 0561, the number of terms is 7, and the common ratio is
S; what is the other extreme'? Jnx. G5Glj3'~'=U.
This rule does not require any exphvnatiou.
EXERCISES.
36. The common ration is 3, tlio number of terms is 7,
and one extreme is 9 ; what is the other ? Ans. 0561.
37. The common ratio is 4 the number of terms is
6, and one extreme is 1000 ; what is the other ? Ans.
1024000.
38. The common ratio is 8, the number of terms i**
10, and one extreme is 402653184 ; what is the other ?
Ans. 3.
In progression, as in many othe rules, the application of
algebra to the reasoning '.voukl greatly simplify it.
MISCELLANEOUS EXERCISES IN PROGRESSION.
1. The clocks in Venice, and some other places stiilvO
the 24 hours, not beginning again, as ours do, after 12.
How many strokes do tliey give in a day ? Ans. 300.
2 A butcher bought J 00 sheep; for the first ho
gav(' 16". , and for the larit i.'9 lO.:;. What did ho pny for
Of
Wl
all, suppi
Ans. £d
3. A
yard he
price of ;
4. A]
the first
on, until
did he tr
5. A
that the
and that
year. I
6. Fii
Ans. '
7.
8.
payment
being £
common
the ratio
9. Wl
thing fo:
second,
shoe .''
10. A
queathec
gave £1
next, li
was the
of the I
ceived i
1. W
series ?
2. W
trical pi
names .^
3. W
ratio ? I
PROGRESSION.
339
nil, supposiig tlicir prices to form an arithmetical scries ?
Ans. iioOO.
8. A person bought 17 yards of cloth ; for the first
yard he gave 2.?., and for the last IOj. What was the
price of all ? Ans. £r> 2s.
4. A person travelling into the country went 3 miles
the first day, 8 miles the second, 13 the third, and so
on, until ho went 58 miles in one day. How many daya
did he travel? Aois. 12.
5. A man being asked how many sons he had, said
that the youngest was 4 years old, and the eldest 32,
and that he had added one to his ftimily every fourth
year. How many had he ? Ans. 8.
6. Find the sum of an infinite series, J, ^j gVj &c.
Ans. 1.
7. Of what value is the decimal 463' ? Ans. ^f f .
8. What debt can be discharged in a year by montlily
payments in geometrical progression, the first term
})eing jei, and the last £2048; and what will be tho
common ratio ? Ans. The debt will be £4095 ; and
the ratio 2.
9. What will be the price of a horse sold for 1 far
thing for the first nail in his shoes, 2 farthings for the
second, 4 for the third, &c., allowing 8 nails in each
shoe ? Ans. £4473924 55. 3frZ.
10. A nobleman dying left 11 sons, to whom he be
queathed his property m follows ; to the youngest he
gave £1024; to the next, as much and a half; to tho
iiext, 11 of the preceding son's share ; and so on. What
was the eldest sou's fortune ; and what was the amount
of the nobleman's property r Ans. The eldest son re
ceived £59049, and the father was worth £175099.
QUESTIONS.
1. Wliat is meant by ascending and descending
series ? [1].
2. What is meant by an arithmetical and geome
trical progression ; and are they designated by any other
names ? [2 and 3] .
3. What are the common difierence and common
ratio } [2 and 3] .
i
II
540
ANNf ITIKS
4 t\).\>^ that a onntiimofl proportion mny 1)0 fomied
froTi, a 8crics of ehhov kiiul r [2 iind 3J.
.'j. Wliat arc moans ii"/i (^xtrcniej ? [4].
6', l^ow is the hm, aritniuetical or a gcomo*
trical series fouiivl ? jo auu Itij.
7. [low is the common difference or common ratio
found ? [ 7 and 1 8] .
8. How is any nimiber of aritliraetioal or geometrical
means fouwd ? [9 and 20] .
9. How is «ny particular arithmclicul or geometrical
mean found ? [10 and 21].
10. How is tlio number of terms in an arithmetical
or geoinctriual series found .? [12 and 23].
11. How is one i^xtremo of an arithnuitical or geome
trical scries found } [14 and 24].
ANNUITIES.
25. An annuity is an income to he paid at stated
times, yearly, halfye;irly, &o. It is either in possession ^
that is, entered upon alread}'', or to he entered upon
immediately ; or it is in reversion^ that is, not to com
mence until after some period, or after something has
occurred. An annuity is certain when its commence
ment and termiuation are assigned to definite periods,
conlingcMt when its l)Oginning, or end, or both are
uncertain ; is in arrears when one, or more payments
are retained after they have become due. The amount
of an annuity is the t.im of the payments forborne (in
arrears), and the interest duo upon them.
When an annuity is paid off at once, the price given
for it is called its ]nescnl: worthy or value — which ought
to be such as would if left at compound interest until
the annuity ceases — 'produce a sum fqual to what would
be due from the annuity left unpaid until that time.
This value is said to l;»^e so many years'* purchase ; that
is, so many annual payments of the income as would "be
just equivalent to it.
26. To find the amount of a certain number of pay
ments in arrears, and the interest due on them —
iJlJT.F..—
the sum of
bo the vci[\
EXAMPI.K
uuiialdfor C
The Inst,
them, form
4 . . XlX^^
Example
is unpaid 1
por emit, p
In this c
with its in
luulli plied
with its ill
Ik the %\\i
''i'lie unu
tlio trouti^■
the series
X2100U2)
• ~1'(
The am
same aa t
years, whi
sov, is eq
Xl; that
the roquii
It vou
ar.nuiiV !
2527 
to bo CO 1
Hence
payment
them —
Subtr
number
interest
tirnt by
ANNUITIES
liUT.F..
341
llion
aud interest duo on tlicm, will
Find the interest dtie on each payment
payinoi
anioun
uin,
Iho smii of the p
Ic tho reiiUKtrd
Faampi.k 1.— Wliat will he the amount of XI .per ann
ui mafd for G years, 5 per cent, simple interest heing allms .dT
The last, and , ^oeedingpaymeni.^vith the inteij^s^^
theui, form tho .'. ^tfnnetical sov.e. ^'^.^''^^^^^^
4 XlxXOS XI. Audits sura 18 X14X1+^^'>X'X
=X24X'25X3=XG'75=X0 15.s., the required amount.
ExAMi'LK 2.If the rent of a {\irm worth XGO r;;^ a.nnuni
is unpaid fo^ 10 year., how much doen u amount to, at .
pt>r cent, per an. compound mtercst .
In thi« ease the series is g.o,..fr/ra/ ; and the last my^^en
wilh its interest is the onunmt of XI for IH (IJ — i) J**''^"
mvlli hod lY tho siven annuity, the pi;eeoding paynie
with itH iutorvst is Iheamomit of Xi fnr 17 years mulUphed
hy the ^^iven annuiiy, &:c. ^^^^ ^^^^^ ^,f
t.^';vc:rTf^l^yiar:^:e2.^^^^^ Then tl. sum of
the series is —
££i2!^^X^:2i^55ll'^l?[l()]=:l •i24, the required amount.
The iMivmnt of XI f.a 18 years mvdtii.lied hy 105 is the
same as the Huu.imt ol XI toi U, oi ilo ol'' , the divi
vo'v^ which is found to he X252/ . And lOo — 1, trie uuj
^;^is e ual to the amount of Cl for oue payment imnua
11'; thltis,tothe Intcre t of XL fc.; one payment. Hence
X252( X <^3i^' _ £18o24.
the required sum will he  " .q^
It would evidently ho th. same ^hin^ to consider^ the
ar.nui;y as XI, and then multiply the re;. It hy 00. lluu
'■^lXni X G0 = X18"'24. For an annuity of XGO ought
to ho GO times as productive as one of only XL
• Hence, briefly, 1 find tlie amount of any nun.her of
ptiynients in arrears, and the compoum intore.t dae on
^^'sl^aet £1 from the amount of ^1 for the giverj
T,mph<u of pavments, and divide the diilcrenco^by 4Jiq
interest of £1 for one payment ; then um.lU^^)iy ine 'i">^.^
tient l>y the "fiyeii nni.
I
^1
t.
w
' ■ ANN urn KS.
re?J*«?ij''.?H'' "*' 'V.1^I*^^^^«— E'icl» payment, with its Into
mn ; K I ^ co„s unto u H.juirnt, a.uu.mt ; „u I tlu, Hu.n dtte
•» u<t bo the sum of tlie.vo aruounlMwliiel, Vunu a ,/.«r Wn^
•ones, because of tlio deoreash.g interest, ari" ng fro m thf
.lecreiw.ug number of times of pa^monl, ^ ®
which 1 nfn « L " °^ ? 7'Mm./,ra/ series, one extreme of
Which IS tiiehrst payment plus tlio interest due upon it at th«
aitterence the interest on one payment due at the next.
Hut when cowpounil interoat is allowed, what is due will hn
the sum <.f a geoniHrical series, one extreme of wEx^sth«
hrst payment plus the interest duo on it at tho a^t the otlor
Jnr/hf •*f^"'?i'/"^* it« common ratio £ pi i'ts tter hI
inteieat due on the hrat payment at the time of the last will
bo the intoros due for one less than tho numLr of payments
payment''"' '' "'' '"^ "° "'° '^«' "^^^^ '''' '^^'^ KoS
EXERCISES.
fn.^i^''''^' '" *^, '^^^'^'^"^^ «f ^'37 per unnu.u unpaid
Ls ^llTios ^'' """*• ^'' """• '''"^^' ^°^"^'^'^
2. What is tho amount of m annuity of i^loo to
continue 5 yeai;s at 6 per cent, per an. conipouua inte
rest .'^ Ans.£bmUs.2\d. i i i^
3. What is the amount of an annuity of ^£350 to
4 What is the amount of ,£49 per annum unpaid
Ans ^ilu 5 ^^^ ''x''^' ^^^'P^'^"^ ^"*^^est b^^i"g 'lUovved .?
28. To find the present value of an annuity—
Rule.— Find (by the last rule) the amount of tho
given annuity if not paid up to the time it will cease.
1 hen ascertain how often this sum contains the amount
ot J^l up to the same time, at the interest allowed.
Example.— What is the present worth of an annuity of
S^^^uf^'itm: 18 ye,.s would amount t,
ANNIHTCKH.
313
But. .CI put to intorost for 18 years at the same ruto
would fimiaiut to X240GG12. Tliorolmo
— *''40(j(?>~" ~ ^® required value.
The sura to be puiil for the annuity should evidently bo euch
ns would produce the aiimo as the annuity itself, ia the same
tima.
liXEKCISES.
f). What is the present worth of an annuity of £27,
to be paid for 13 years, o per cent. coinpoTind interest
being allowed ? Ans. £2j3 \2s. (j\d.
6. What is the present worth of an annuity of J6324,
to bo paid for 12 years, 5 per cent, compound interest
beiug allowed? Aiis. £2671 I3s. 10}f/.
7. What is the present worth of an annuity of .£22,
to be paid for 21 years, 4 per cent, compound interest
being allowed } Ans. £308 12s. lOd,
29. To lind the present value, when the annuity is
in perpetuity —
lluLK. — l)ivide the interest which £1 would produce
in perpetuity into £1, and the quotient will be tno sum
reijuirud to produce an annuity of £1 per annum in
perpetuity. iMultiply the (uutioiit by the number of
pounds in the given annuity, and the product AtII be
the required present worth.
Example. — Wliat is the value of an income of £17 for ever *
Let us suppose that XlOO would produce £b per cent, per
an. for ever: — thou £i would produce £05. Therefore,
to produce £1, we roquire as many pounds as will be equal
to the number of times £05 i,<* contained in £1. But7r?=ai
£20, therefore £20 Avould produce an annuity of £1 for
ever. And 17 times as much, or £20x17=340, which
would produce an annuity of £17 for over, is the required
present value.
EXERCISES.
8. A small estate brings £25 per annum ; wh&t is
its present worth, allowing 4 per cent, per .annum irste
rest .? Ans. £62.5.
9. What is the present worth of an incorae of £347
; 1
I
344
ANNUITIES.
in perpetuity, allowing G per cent, interest? Ans
£5783 6s. S(L
10. What is the value of a perpetual annuity of £46,
aUowiug 5 per cent, interest ? Arts. ^£920.
_ 30. To find the present value of an annuity in rever
sion —
lluLE. — Find the amount of the annuity as if it were
forborne^ until it should cease. Then fi^dwhat sum,
put to interest now, would at that time produce the
same amount.
Example.— What is the value of an annuity of £10 per
annum, to continue for 6, but not to commenr for 12 years,
o per cent, compound interest being allowed ?
^'^1 ""J^^ofe^ ^^^ ^'^^ ^ ye^^'« if left unpaid, would bo
l'?'!^u^a'^]^} ' ^^'1 ^1 ^0"!^' i» 1^ years, be worth
i^ii'ObUoO. Ihercforo
M80VJI
ll~08959~^"^ ^^' '^'■^•■> i^ ^^^^ required present worth.
EXERCISES
11. what is the present worth of .£75 per annum,
which is not to commence for 10 y^ars, but will con
tinue 7 years after, at 6 per cent, compound interest ?
Am. £1.55 9.?. 7^d.
12. The reversion of an annuity of £175 per annum,
to continue 11 years, and commence 9 years hence, is to
be sold ; what is its present worth, allowing 6 per cent,
per annu.n compound interest ? Ans. £430 7*. }d.
13. What is the p^esent worth of a rent of £45 per
annum, to commence in 8, and last for 12 years, 6 per
cent, compound interest, payable halfyearly, being
allowed.' ^?w. £117 25. S^^.
31 When the annuity is contingent, its value depend.^
on the probability of the contingent circumstance, or
circumstances.
A life annuity is equal to its amount multiplied by '
the value of an annuity of £1 (found by tables) for tho
given age. ^ The tables used for the purpose are calcu
l:i ted on principles derived from the doctrine of chances,
observations on the duration of life in different circum
stances, the rates of compound interest, &c.
POSITION.
345
QUESTIONS.
1 . What is fin annuity ? [25'j .
2. What is au annuity in possession — in reversion —
certain — contingent — or in arrears ? [25].
3. What is meant hy the present worth of an an
nuity ? [25] .
4. Plow is the amount of any number of payments
in arrears found, the interest allowed being simple or
■compound ? [26] .
5. How is the present value of an annuity in posses
sion foimd ? [28].
6. How is the present value of an annuity in per
petuity found ? [29] .
7. IIow is the present value of an annuity in rever
sion found .' [30].
^'
i 1
POSITION.
32. Position, called also the " rule of false," is a rule
which, by the use of one or more assumed, but false
numbers, enables us to find the true one. By means of
it we can obtain the answers to certain questions, which
we could not resolve by the ordinary direct rules.
When the results arc really proportional to the sup
pof'ition — as, for instance, when the number sought is
to be muUijflied or divided by some pi'oposed number ;
or is to bo increased or diminished by itseJf, or by some
given mnUijplc. or j)art of itself — and when the ({uestion
contains only one p)opositio7ij we use what is called
single position, assuming only one number ; and tho
(juantity found is exactly that which is required. Other
wise — as, for instance, when the number sought is to bo
increased or diminished by some absolute number, which
is not a knovrn multiple, or part of it — or when two
propositions, neither of which can be banished, are con
tained iu the problem, we use douhk position, assuming
itjuo numbers. If the number souiclit is, durins; tho
process indicated by the question, to be involved or
evolved, we obtain only au approximation to the quan
tity required.
I
Ill
346
POSITION.
.'«. ^^ingk PosiUon.~B.vhK. Assume a number, and
perform with it the operations described in the question ;
then say, as the result obtained is to the number nse(l\
so IS the true or given result to the number required.
ExAMPLK.— What number is that whicli, being multiplied
^J ». '^y i, and by 9, the sum of the results shall be 231 ?
Let us assume 4 as the quantity sought. 4x5j4x74
4x0=84. And 84 : 4 :: 231 : ^^ll, the required
number.
84. Reasoiv of the Rule.— It is evident that two num
bers, mnlUpaed or divided by the same, should produce pro
poi t.onatc results It is otherwise, ho^yever, when tJie ^me
quantity is added to, or subtracted froia tliem. Thus let tho
given question be changed into the following. What number
IS that wiuch being multiplied by 5, by 7, and by 9, the sum
ol the products, plus 8, shall bo equal to 239 .>
Assuming 4, the result will be 92. Then we cannot say
92 (84+8) : 4 : : 239 (2318) : 11.
For though 84 : 4 : : 231 : 11, it does not follow that
h4j8 : 4 :: 2olj8 : 11. Since, while [Sec. V. 29] we may
multiply or divide the first and third terms of a geometrical
proponiun by the same number, we cannot, without destroy
ing the proportion, add the same number to, or subtract it
trom thorn. The question in this latt«r form belongs to tho
rule of duuble pcsitiou.
EXERCISES.
1. A teacher being asked how many pupils he had^
replied, if you add l, ] , and J of the number together,
the sum will be IS ; what was their number > Ans. 24.
2. What number is it, which, being increased by \,
J , and i of itself, shall be 125 } Ans. 60.
3. A gentlcnuin distributed 78 pence among a num
ber of poor persons, consisting of men, women, and chil
dren ; to each man he gave GiL^ to each woman, 4rZ.,
and to each child, 2d. ; there were twice as many
women as men, and three times as many children as
women. How many were there of each } Ans. 3 men,
f^ women, and IS children.
^ 4. A person bought a chaise, horse, and harness, for
£t){) ; tbe horse came to twice the price of the harness,
and the chaise to twice the price of the horse and har
POSITION.
347
flCKS. "What did he give for each ? Am. He gave for
ihc harness, £C) I'Ss. 4(1. ; for the horse, i^^lS ijs. Sd. ;
uud for the chaise, iD'lO.
5. A's age is double that of J5's ; IVs is trchlo that
of C's ; and the sum of all their ages is 140. What is
the age of each ? Ans. A's is S4, J3's 42, and C's 14.
6. After paying away J of my money, and then } of
the remainder, I had 72 guineas left. What had I at
Krst ? yh^s. 120 guineas.
7. A can do a piece of work in 7 days ; J* can do the
same in 5 days ; and C in G days. In what time will
ull of them execute it ? Ans. in Ij^^ days.
8. A and B can do a piece of work in 10 days ; A
by himself can do it in 15 days. In what time will 13
do it ? Ans\ In 30 days.
9. A cistern has three cocks ; when the first is opened
all the water runs out in one hour ; when the second is
opened, it runs out in two hours ; and when tlic tlurd i?
opened, in three hours. In what time Avill it run out, it'
ail the cocks are kept open together r Ans. In /y hours.
10. What is that number whose ■}, J, and } parts,
taken together, make 27 ? Ans. 42.
11. There are 5 mills; the first grinds 7 bushels of
corn in 1 hour, the second 5 in the same time, the third
4, the fourth 3, and the fifth 1. In what time will the
five grind 500 bushels, if they work togctlier ? A71S.
In 25 hours.
12. There is a cistern which can be filled by a cock
in 12 hours ; it has another cock in the bottom, by
which it can be emptied in IS hours. In what time will
it be filled, if both are left open ? Ans. In 36 hours,
35. Doiibh Position . — Rule I. Assume two con
venient numbers, and pe.'form upon them the processes
supposed by the question, marking the error derived
from each with + or — , according as it is an error of
e.iwvs, ©r of defect. Multiply each assumed number into
the error which belongs to the other ; and, if the errors
are hot/i plus, or hoth minus, divide the diJJ'crence of the
products by tho difference, of the errors. ]3ut, if one is
a plus, and the other is a minus error, divide the sum of
I M
348
POSITION.
the products by tlio mm of the errors. I,, either cas(^
the result will bo the nuiriber 60U'.lit, or an apuroxi
iiiatiuu to it. ^ ■
Exami'm: 1.— If t„ 4 timpg the price of my horse XIO U
added, the sum will be £100. AVliat did it cast '?
As.suming muubers wliich give two errors oi" excess^
First, lot 28 be one of thoni,
Multiply by 4
112
Add 10
From 122, the result obtained,
subtract 100, the result required,
and the remainder, +22, is an error of crrew.
iMultiply by 31, the other assumsd uuinlwr
and 082 will bo the product.
Next,, let the assumed number be 31
Multiply by i
124
Add 10
From 134, the result obtained,
subtract 100, the result required,
, and the remainder, {34, is an error o?nrrc'^<i.
Multiply by 28, the other assumed uum,
and 'J52 will bo tlic produot.
From this subtract G82, thoproductfouud.diove,
divide by 12)270
and the required quantity br22r)=r£22 10s.
Difference of crrora=34.':;^12, the immber h\ whieli
._. .lift iiiimhdv ]\x «']ii,.1
we have divided.
_ 36. Rkason of thk ?vUle.— AVhen in cxrimnle 1, we mul
tiply 28 and 81 by 4, weiuuitiply the error belongini to e.icli
by 4. Hence 122 and 134 are, reHpecLivelj, cquuf to tlie true
result, plus 4 times one of the error.s. Subti;ictiu<r tO(3, tb<j
true result, from e;icli of them, we obtahi 22 (4 tiiue.ri >e errur
in 28) and iJ4 (4 times tlie errur in 81).
But, as numbers are propurtioniil to their '^y?//miltiples
the error in 28 : the error in 31 : : 22 (a multiple of the for
mer) : 84 (au enuinuiltiple of the latter).
And from the nature of proportion [Sec. V. 21]—
POSITION.
349
The error in^28x34==thejBiwrM^^
Cut 682= tiio~error in"31f t)ie required number X22.
And 95'2=the error in 28tiie required uuiabevX34.
Or, since to multiply quantities under the vinculum [Sco
[I. 84], we are to multiply each of them— ■
t)82=22 times the error in 3if22 times the required number.
it52=34 times the error in2843i times the vequiied number.
Subtracting the upper from the lower line, we shall have
952—682=34 times the error in 28—22 times the error in
31484 times the required numbcv— 22 times the required
number. • oq o i
But, as we have seen above, 34 times the error in jy=_'.i
times the error in 31. Therefore, 34 time:s the error in 28—22
times the error in 31=0; that is, the two quantities cancel
each other, and may be omitted. We shall then have
952 — 082=34 times the required number— 22 times the re
quired number; or 270=3422 (=12) times the required
number. And, [Sec. V. G] dividing both the equal quanti
ties by 12,
OTA 34—22
"in (22 '5) =  T^ *i"^C'* (once) the required number.
37. ExAiNiPi.E 2.— Using tho same example, and assivming
nmnbora which ^ixa two errors of defect.
Let them be 14, and 3.0 —
14 16
4 4
50
10
GO, the rosnlfc obtained,
100, the result required,
 34, an error of defect.
10
04
10
74, tiie result ol)tainod,
100, the resuk required,
— 20, an error of defect.
14
544
304
304
Difference of errors :
III
•I
: 34 — 20 = 8.
8)180
225 =£22 lOs., is the required quantity.
In this example 34=.four times the error (of defect) in 14;
and 26= four times the error (of detect) in IG. And, yineo
ftiumbevs are pvoportioptd to iJieir equimultiples,
The error in 14 : ih  error in IG : : 34 : 2(5. Therefore
The error in ! lx2G=:tho error i u 16 X34.
But 544=the required number — the error in 16X81
And3Gl=thc required number — the error in MX2G
«i 2
1
POSITIOX.
If wc subtract tho lower from tlic upper lino, we shall Imvo
644 — o(>4=(rcinuviiig the vinculuia, uud changing tho «igu
[Sec. 11. It)]) ;M times the requiriMl number— 2(J times tTio
lequired iiuuiber — ol liiiuis the civui in 1G4'J times tho error
ill 14.
IJut we found above that 34 times the error in 16='26 times
tlie error in 14. Therefore — 34 times tho error in l<j, nnd4"<>
times tlie error in 14=i0, and niiiy be onntted. We will then
have 544 — ;J(>1=34 times the required number — 2t> times the
required number; or 180=8 time^ the required number; and,
dividing botli these equal quantities by 8,
180 8
— Q (22'5) =,T times (once) tlie requtrcd number.
38. ExAMi'LE 3. — Using still the same example, and as
sinnlng numbers "vvhich will give an error of extaw, and an
error of defect.
Let thorn be 15, and 23 
15
4
60
10
70, tho rosiult ohto.intMl.
100, tho result required.
 SO, an error of defect.
23
C'JO
30
23
4
92
10
102, tho result obtained.
100, tho result rcquirod.
2, an err 01' of exie^s.
15
30
Sum of errors = 30 f 2 :
32.
32)720
225 = £22 lO.s., the required quantity.
In this example 30 is 4 times the error (of defect) in 15;
and 2, 4 times the error (of excess) in 23. And, since numbers
are proportioned to the equimultiples,
The error in 23 : the error in 15 : : 2 : 30. Therefore
The error in 23X30=the error in 15X2.
But l)UO=::the required iiumberfthe error in 23x30.
And 30=the required number — the error in 15X2.
If Ave add these two linos together, we sliall have b!!0f 30=
(removi;ig tlie vinculum) 30 times the required numbor+
tvvice the required number {30 times the error in 23 — twice
ihe error in 15.
But we found above tliat 30 X the error in 23r=2v;;io error
in 15. Therefore 30!i^the error in 23 2 X the 'irror in 15=0.
I'O'MTIO:
351
,
S . w';*'>> t.^ A'ud divldiug each «C tl..e equal
quantities by 32.
'^20,<,2.6^= times (once) the required number.
8 ' oit
The given questions might be changed into one belongmg
to sin^'/e position, thus—
p„u. times the price f "/.hj^'i =» °X l90.''"wlaf ..id
or four times the price of "X ''<'''•« '''"j" „„ ^frort of the mind
39. ExAMPLK 4.What is that number which is equal to
4 times its square root +21 '?
Assume 64 on'' 81
4
n2
21
^81= 9
4
36
21
53, result obtainc'l.
G4, result ro<iuired.
81
57, result obtained
81, result required
64
1536
891
13)645
The first approximation ia 49'6154
U is evident thatU and 24^ ^Z'lIX^ '^^^
„,„„bcrs multiplied or '^"'^'"LteuSe rule is founded, docs
therefore, as the reason "P°" "™„o"hnation. Substituting
?,^iV >S;;^Wrfo?r or^assSeTnuntbers. we ohtatn .
(Btill nearer approximation.
TT T7m,l the errors by the last rule ; then
kind), or their sum ^f they are o^J^"J^ ^ ^^^e of
error which has been used as multiplier.
352
rosiTiox.
KxAMrr.K. — Taking the Hamo an in tlic lust rule, and a/
Burning I'J and 23 as tho reiiuired numbor.
19 25
4 4
70
10
8(5 the result obtained.
100 the result required.
—14, is error oi defect.
110 the result o))tainfMl.
loo the result required.
f10, ia error of excels.
The errors are of different Y\x\({^; and their sum is 14f
ro=24 j and tho difference of the assumed numbers is 25 —
1U=(}. Therefore
14 one of the errors,
is multiplied by 0, by the difference of the numbers. Then
divide by 24)84
and 35 is the correction for 19, the number
which gave an error of 14.
194(the error being one of defcpt, the correction is to bo
added) 3 5=22 5=£:i2 IDs. is tho required quantity.
41. Reason of the Rule. — Tlie diifercnce of tho results
arising from tlie use of the different assumed numbers (tho
difference of the errors) : the difference between tho result ob
tained b^'' using one of the assumed numbers and tliat obtained
by using the true number (one of tlie errors) : : the difference
between the numbers in the former case (the difference betweea
the a.sHumed numbors) : the difference between thu numbei'S
. in the latter case (tlie difference between the true mmiber, and
that arjsumed number wiiich produced tlie erior placed in the
thir<l term — tliat is the correction required by that assumed
number).
It is clear that the difference between the numbers used
produces a proportional difference in the results. For tlie
results are different, only because tho difference between the
assumed numbers has been multipliccl, or divided, or both —
iu acconlance with tho conditions of the question. Thus, in
the present iuytsmce, 25 pl'oduoes a greater result thau 19,
because 0, the difference becween 19 and 25, has been multi
plied by 4. For 25x4=s=19x4f6x4. And it is this 6X4
which makes up 24, the rtal difference of tlie errors. — The
difference between a negative and positive result being the
sum of the differences between each of them and no result.
Tbus, if I gain 10s., 1 am richer to the amount of 24*. than if
1 lose li.*.
^
rosiTiuN.
353
ilo, and M
t o))tainfMl.
fc required.
of execs'^.
im is 14+
era is 25 —
ors. Then
do number
on 13
to bo
the results
iiubers (tlio
3 residt ob
lat obtained
e difference
ace between
\ii numbers
lumber, and
aced in the
at assumed
mbpvs used
s. For tlie
)etween the
, or both —
. Tims, in
;lt thau 19,
been multi
is this 6x4
rrors. — The
. being the
I no result.
24«. than if
t
EXEUCISES.
13. What number is it whit;li, boiur; niultipliod by 3,
Uio product \n\\\\% increasod by 4, and the sum divided
by 8, the quotieut will bo \V2 ? Ans. 84.
14. A4!ron asked liis fatlier how old lie was, and re
ceived the foUowin,]; answer. Your ago is now J of
What are their
mine, but 5 years ago it was only i.
ag(\s } Ans. 8;) and 20
' 15. A workman was hired for 30 days at 2s. Qd. for
every day ho worked, but with this condition, thafc^ for
every day he did not work, he should forfeit a shilling.
At the end of the time he received £,2 14^., how many
days did he work ? Am. 24.
16. llcquired what number it is from which, if 34
be taken, 3 times the remainder will exceed it by \ o^
itself .= Ans. 58=. .
17. A and 13 go out of a town by the same road. A
g(jcis 8 miles each day ; 15 goes 1 mile the fir.st day,
2 the second, 3 ' ' ' '
take A }
the third, &c. When will B over
Suppose
A.
5
8
40
15
B.
1
2
3
5
Suppose
A.
7
8
50
28
5)25 Is
a
7
35
20
1)15
7)28
4
5
20
B.
1
2
• 1
O
4
5
G
7
28
5
4=1
"Wc divide tho ciilive eri'nr by the number of daya iu each
c..sc, which gives tlie error iu one day.
18. A gentleman hires two labourers; to the one bo
gives M. each day; to the other, on the first day, 2(/.,
on the second day, Ad.^ on the third d;iy, 6^/., &.c. In
how many days will they earn :tn e'pinl sum } Aim. In 8.
11). What" are tho.s:j numhers whieh, when added,
ij
S54
I'OSrTION.
make 25 ; but when ono in halved and the other douhled,
give (H[na[ results ? Ans. 20 aud 5.
20. Two coutractoiH, A and ]J, arc each to hiiild a
wall of equal dimcusioiis ; A (iniploys as many men as
finish 22J perches hi a r'ay ; Ij employs the first day as
many as finish G perches, the second as many as finish
9, tlie third as many as finish 12, &c. In what time
will they have built an e(ual number of perches ?
Ans. In 12 days.
21. What is vhat number whose ^, i, and }, multi
plied together, make 24 ?
Sujtposo 12
1=3
rroduct=i8
3 41
81 result obtained.
21 result rc(iuircd.
+57
04, the cube of 4.
3648, product.
57+21=78
Suppose 4
Product=r;2
a — U
3 result obtained.
24 result required.
1728, the cube of 12.
30288 To this pruduct
3018 is added.
5<
721=78.
78 )391)3 is the sum.
And 512 tlio quotient.
3/512=8, is the required number.
We nmltiply the alternate error by the cube of tlie supposed
tmmbcr, because tlui errors belong to the g'^th part of the cube
of tlie assumed numbers, and not to tlic nuinbei'S tlicniselvos ;
for, in reality, it is the cube of some number that is required
—tt'ince, 8 being hssumed, according to the question we have
22. What number is it whose 1, J, ], and 1, multi
plied together, will produce 699S .'^ Am. 36.^
23. A said to B, give me one of your shillings, and
I shall have twice as many as you will have left. B
answered, if you give me Is., I shall have as many as
you. Ilow ninny bird each } Avs. A 7, and B 5.
POSITION.
355
24. There are two nuinbors wliich, when ailuo(1 to*
gather, iiialca 30; but the J, J, and j, of the greater
arv cqijal to , a^ and ^, of the lesser. What are they ?
Ans. Vj and Ijs.
2' A f^ontlomaii has 2 horsoa and a sacMlo worth
j£50. Tbo ,s:iddlo, it' set ' n th* baek of the first linrse,
will make his value doubic that of the second ; but if
set on the baek of the second horse, it will make his
value treble that of the first. AVhat is the value of
each horse ? j v. £30 and iJlO.
2C>. A gentleman finding sov ral ben;£^ars at his door,
pavt! to eii;h 4d. and had Gd. left, but if he had given
ikl. to each, he would have had 12d. too little. How
many bop:gars were thure ? Aiis. 9.
It is so likely tliat those ) are desirous of stud^inr;
this subject further will be acquainted with the method
of troatin<5 algebraic equations — which in miiiy case?
nffords a so much simpler and easier mode of solvin,';
qu.;stions belonging to position — that we do not deem
it necessary to enter further into it.
QUESTrONS.
1. What is the diiTerence between single and double
position.? [32].
2. In what cases may we expect an exact answer by
ihesc rules r [32 j .
3. Whtit is the rule for sin<>le position ? [33] .
4. AVhat are the rules for double position .'* [35 and
40 j.
MISCELLANEOUS EXERCLSES.
1. A father being disked by his son how old he was;
voplied, your age is now ^ of mine ; but 4 years ago
\i was only ^ of what mine is now ; what is the ago of
each } A as. 70 and 14.
2. Find two numlxns, the dilForence of which is 30,
nnd the relation between them as 7^ is to 3^.'' Am.
58 and 28.
3. Find two numbers whose sum and product are
equal, neither of (hem being 2 ' Ans. 10 and 1^.
.%
o.. \t^
IMAGE EVALUATION
TEST TARGET (MT3)
/.
^° 'C^x
 ^> /^^^ j/^
C^,
:/.
%
1.0
I.I
11.25
1.4
IM
[2.0
1.6
PhotograDhic
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, NY. 14580
(716) 8724503
,,y^':
o
356
EXERCISES.
What is the t;um of tlie series J, i, }, &c. ? Ans. 1.
4. A^porson being asked the hour of tlio day, answered,
It is between 5 and G, and 1;otli the hour and minuto
hands are together. lle(iuired what it was ? Ana.
27 f\ minutes past 5.
5.
6. What is the sum of the .series' ,'y\, j%y j'A) ^^ •
Ans. If
7. A person had a salary of £75 a year, and let it
remain unpaid for 17 years. How much had he to
receive at the end of that time, allowing 6 per cent,
per annum compound mtercst, payable halfyearly ?
Ans. £204 17s. lO^d.
8. Divide 20 into two such pav^s as that, when tho"
greater is divided by the less, and the loss by tlie greater,
and the greater quotient is multiplied ))y 4, and the less
by 64, the products shall be equal.? Ans. 4 and IG.
9. Divide 21 into two such parts, as that when the
less is divided by the greater, and the greater by the
loi'S, and the greater quotient is multiplied by 5, and
ih" less by 125, the products shall be e(pial ? Ans.
3' and 171.
1" A, B, and C, can finish a piece of work in 10
days; }> and C will do it in 16 days. In what time will
A do it by himself? Ans. 26 days.
1. A can trench a garden in 10 days, B in 12, and
in 14. In what time will it be done by the three if
thoy work together ? Ams. In 3,Yt ^'^J^
12. What number is it which, divided by 16, will
leave 3 ; but which, divided by 9, will leave 4 ? Aiis.
67
i3. What number is it which, divided by 7, will
leave 4; but divided by 4, will leave 2 ? Ans. IS.
14. If £100, put to interest at a certain rate, wih,
at the end of 3 years, be augmented to £115'7G25
(compound hitercst being allowed), what principal and
interest will hi due at the end of the first year ? Ans.
£105.
15. An elderly person in trade, desirous of a little
respite, pi'oposcs to admit a sober, and industrious young
person to a share in the business ; and to encourage
him, ho olfors, that if hi^ circumstances allow him to
F,XERCrSE!5.
357
advance £100, his salary shall be £40 a year ; that if
ho is able to advance £200, he shall have £55 ; but
that it he can advance £300, he shall receive £70
annually. In this proposal, what was allowed for his
attendance simply ? Am. £25 a year.
16. If 6 apples and 7 pears cost 33 pence, and 10
apples and 8 poars 44 pence, what is the price of one
apple and one pear .?* Ans. 2d. is the price of an apple,
and 3d. of a pear.
17. Find three such numbers as that the first and I
the sum of the other two, the second and i the sum of
the other two, the third and \ the sum of the other
two will make 34 ? A^is. 10, 22, 26.
18. Find a number, to which, if you add 1, the sum
will be d"visible by 3 ; but if you add 3, the sum will
be divLsiVie by 4 ^ Am. 17.
19. A market woman bought a certain number of
eggs, at two a penny, and r,s many more at 3 a penny ;
and having sold them all at the rate of five for 2^., she
found she had lost fourpence. How many eggs did she
buy .? Am. 240.
20. A person was desirous of giving 3d. a piece to
some beggars, but found he had 8^. too little ; he there
fore gave each of them 2d., and had then Sd. remain
llequired the number of beggars.? Am. 11.
21. A servant agreed to live with his master for £8
a year, and a suit of clothes. But being turned out
at the end of 7 months, he received only £2 135. 4d.
and the suit of clothes ; what was its value } Am,
_ 16.9.
22. There is a number, consisting of two places of
figures, which is equal to four times the sum of its
dtgits, and if 18 be added to it, its digits will be in
verted. Wiiat is the number ? Am. 24.
23. Divide the number 10 into three such parts, that
if the first is multiplied by 2, the second by 3, and the
third by 4, the three products will be equal .? Am.
24. Divide the number 90 into four such parts that,
If the first i;^ increased by 2, the second dhninished by
2, the third multipli(!d by 2, and the fourth divided by
mg
358
EXERCISES.
%
2, the sum, clIfForcncc, product, and quotient will bo
equal : Ans. 18, 22, 10, 40.
25. \Vliat fraction is that, to the numerator of which,
if ] is added, its viluo will be i ; but if 1 bo added to
tHe denominator, its value will be •} ? Ans. j%.
2^3. 21 gallons were drawn out of a cask of wine,
which had leaked away a third part, and the cask
being then guaged, was found to be half full. How
much did it hold ? Ans. 126 gallons.
27. There is a number, ^ of which, being divided by
6, I of it by 4, and J of 'it by 3, each quotient will
be 9 ? Ans. 108.
28. Having counted my books, I found that when I
multiplied together i, j, and f of their number, the
product was 162000. How many had I ? Ans. 120.
29. Find the sum of the series l+'^f j + , &o. .?
Ans. 2.
30. A can build a wall in 12 days, by getting 2 days'
assistance from B ; and B can build it in 8 days, by
getting 4 days' assistance from A. In what time will
both together build it ? Ans. In 6f days.
31. A and B can perform a pisce of work in 8 days,
when the days are 12 hours long ; A, by himself, can
do it in 12 days, of 16 hours each. In how many days
of 14 liours long will B do it } Ans. 13^.
32. in a mixture of spirits and water,  of the whole
plus 25 gallons was spirits, but i of the whole minus 5.
gallons was water. How many gallons were there of
each } Ans. 85 of spirits, and 35 of water.
33. A person passed } of his age in childhood, yV of
it in youth,  of it +5 years in matrimony ; he had
then a son whom he survived 4 years, and who reached
only i the age of his father. At what age did this per
son die ? Ans. At the age of 84.
34. What number is that whose i ejjcecds its \ by
72 ? Ans. 540. ^ _ .
35. A vintner has a vessel of wine containing 500
gallons ; drawing 50 gallons, he tlicn fills up the cask
with water. After doing this five times, how much
wine and how much water are in the cask.? Ans
295^j)_ gallons of wine, and 204 J i gallons of water.
''
! ,
ni
EXERCISES.
350
45. A mother an<l two daugliters working together
nil 3 lb of fliix in one day ; the mother, by herself,
ian do it in 2i days ; and the eklest daughter m ^j
days. In what time can the youngest do it.? Ans.
In f')— davs.
37^ A merchant loads two vessels, A and B ; into
A he puts 150 hogsheads of wine, and into B 240 hogs
heads. The ships, having to pny toll, A gives 1 hogs
15 <.;ivos 1 hogshead and 3().s\
iich hogshead valued ?
head, and receives V2s.
besides. At how much was c
Ans. £4. 12.S. , ^, .
38. Tlireo merchants traffic in company, and their
stock is i2400 ; the money of A continued in trade 5
months, that of B six months, and that of G nine
months; and they gained £^7b, which they divided
cfpially. What stock did each put in.? Ans. AiilbT^j,
39. A fonntain has 4 cocks, A, B, C, and D,_and
unilcr it stands a cistern, which can be filled by A lu G,
by B in 8, by in 10, and by D in * . hours ; the
cistern has 4 cocks, E, F, 0, and II; and can be
emptied by E in G, by F in 5, by Q in 4, and by II m
3 hours. "Suppose the ci^^tern is full oi water, and that
the S cocks are all open, in what time will it be emptied ?
Ans. In2,^g hours.
40. What is the value of 2^07' ? Ans. if
41 What is the value of 541 G' ? Ans. Yi
42. What is the value of •0^7G923' t Ans. yV
43. There are" three fishermen. A, B, and C, who
have each caught a certain number of fish ; when A's
fish and B'sare put to:rother, they make 110 ; when
B's and CVs are put together, they make 130; and when
A's and C's are put together, they make 120. It the
fish is divided equally among them, what will be each
mairs share; and how many fish did each of them
catch ? Ans. l<lach man had GO lor his share ; A caught
50, B GO, and G 70.
44. There is a golden cup valued at 70 crowns, and
two heaps of crowns. The cup and first heap, are wortli
4 times the value of the second heap ; but the cup and
second heap, aie
worth double the value of the first
3r,o
EXERCISES.
heap. ITow many crowns arc there in eacli lieap ? Ana
oO ill one, and 30 in another.
45. A certain number of horse and foot soldiers ai'O
to be ferried over a river ; and tliey agree to pay 2^d.
for two horse, and Sid. for seven foot soldiers ; seven
foot always followed two horse soldiers ; and when they
were 'dl over, the ferryman received ^£25. How many
horse and foot soldiers were there ? Ans. 2000 horse,
and 7000 foot.
46. The hour and minute hands of a watch arc' to
gether at 12 ; when will they be together again ? Ans.
at 5/, minutes past 1 o'clock.
47. A and 13 are at opposite sides of a wood 135
fathoms in compass. They begin to go round it, in the
Same direction, and a* the same time ; A goes at the
rate of 1 1 ftithoms in 2 minutes, and B at that of 17
in 3 minutes. How many rounds will each make, before
one overtakes the other ? Ans. A wiU go 17, and 13
16J.
48. A, B, and 0, start at the same time, from the
(same point, and in tlie same direction, round an island
73 miles in circumference ; A goes at the rate of 6,
B at the rate of 10, and C at the rate of 16 miles per
day ^ In what time will they be all together again .^
Ans. in 36 days
IX
MATIIEMATICAL TABLES
LOGATUTIIMS OF NUMBERS FllOM 1 TO 10,000, WITH
DIFFKIIENCES AND PROPOllTIONAL I'AllTS.
I '
1
m^^M
Numbers f
•om 1 to 100.
#
I
No. Log. j No.
Log.
NO.
Log.
No.
Log.
No.
Log.
I
1
.
OO'JOOOO
•21
132'2'219
41
1614784
61
1785330
81
1908486
H
■i
0 301030
•22
1342443
44
1043249
64
1792394
84
1913314
1
3
0477141
23
1361743
13
I 0334 .3
63
1709341
83
1919073
4
060JOGO
21
1380211
U
1C43453
64
1800180
34
1924479
1
(i
0 690970
45
'26
1397040
45
46
1053413
05
1814913
35
1929419
!jHH
0'778151
1414973
1664758
66
1819544 66
19^4498
H
0Sl.JOOJ
•27
1431364
47
1674093
67
1346075 87
1 9.39519
^H
8
0903000
•23
1417158
43
1 631241
6S
1834509
83
1944483
mH
9
09J4243
29
1 •462393
49
16901%
69
1838349
89
1949390
■
10
1000000
30
31
1477121
60
51
1693970
70
1845098
90
1954413
■
11
l0413ri3
1491364
1707570
71
1851'253
91
1959041
M
12
1 079181
32
1505150
53
1716003
74
18573.34
92
1963783
^H
13
1113943
33
1518514
63
1724476
73
1863343
93
1 968433
^H
14
lUOlia
31
1531479
54
1732394
74
1869434
94
1973128
iHI
15
1 170091
35
36
1644063
65
56
!■ 740303
1
75
1875061
95
1977724
^H
IG
l2041iO
1550303
1748188
76
1 830314
96
198'2471
m
17
1 230449
37
1533404
57
1 766875
77
1886491
97
1986772
H
13
\2iio2l'i
33
1 •5r9784
53
1763143
73
1894095
93
1991226
■
19
1478754
39
1591005
59
1770854
79
1 SO? 647
99
1995635
■
•2ii
I • 301030 40
1604060
CO
1773151
SO
1 903090
100
2000000
IB^^I
"mm
302
rOGARITIIMP.
r 1'
100
1 1
3 1 3 1 4
6
6
7 1 8
1).
132
:oouuoooooi3i
0008O.sOOI3ni0017.34 UO'2IOO
002.598'00.3029 00.3461
003891
II
11 inJll 4751
6181 660!)! fif38 6466
68941 7321 1 7748
8171
128
Hii
•2 K000[ !)020
3'01'JdH7 013'20!)
9l,.l !)n76 010300 010724
011147i01l570011993;0l211..
121
101
tU30KO 014100 4.V21
4910
6360
5779! 6197
6616
120
Ifiti
I
7033 74')]
7868 8284 8700
9116
9532
9947 020361
020775
116
■'o;
6
02llH!)uai(i03
022ai6]02'2428 022811
023252
023604 024075 4486
4896
112
VilH
G
TMHy 571;,
61 •2..
6533! f:942
7350
77.)7 8164 8571
'3978
108
•J!M)
7
'3' ll 978l'i
03019.)
030(ilU):031004031 108
031812 032216 032619
0M02I
101
;t;il
/24',mi26
42;;7
4628, 6029
5430
6830 62301 6629
702.M
!0l»
;);:)
110
',4:20 7826
8223
H620 9017
9414
9811 040207,040602
040998
;J97
.193
(i;i3:)3 011787
042l82012.)76 042909
043302
043755 044148 044.540
0449.32
;j^
1
;':.323j 6714
OlOril 6I9.)I 688.'.
7275
7064
8053 8442
8830
390
7li
• 1
9218 iUiOli
9993 J0380'n.'.070fi
051153
051538
051924 0.52309
0.52(i91
386
li:i
3
003('7S (l;(3403
053846 4230 461.1
4;)96
5378
,OT60 6112
6521
.■i83
1.^1
4
UDO/)! 72b0
7006 8046 8126
8805
9IH5
9563 9942
060320
.179
iy:<
t
OGO()!I800107')
Oai4.>2 061829 062200 0(!2J82
002958
063333 063709
4083
376
•M^
6
4i;)S 4832
520(1 ooSn 6!»53 6320
6699
7071 1 7443
7815
373
•]«.,
7
81Si)i 8;)i")7
8928 9298 UtiUS 07003S
070407
070776,071115
071611
370
aoo
H
0718+2 07U^2;")0
072617 07298., 0733..2 3718
4085
4151
4816
5182
366
:M()
lyo
.');:.47 .01)12
6276 0640 7004 7308
7731
8094
8457
8819
163
:J60
079181 07l)")43
0799U4;080200'080ll26
080937
081347
081707
082067
08212(;
3;.
1
08278.., 083141
083..03 3861
4219
4576
493 1
5291
5647
6001
357
70
o
0;l!iiH 0710
707 1 712{i
7781
8136
8490
88 15
9198
9552
355
101
3
990;V090208
09061 1]090903
091315 09 I6(i7
092018
092370
092721
093071
:J52
i;i<i
4
093t22J 3772
4122 4471
420
5169
5513
rySGii
6215
656^
.549
171
6
091(1 ~2:u
7604 79.,l
8298
864 I
8990
9335
9681
10002(i
346
20;i
100371' 10071.".
1010;;9 10! 103
101717
10'2091
102434
102777
103119
3162
.113
■211
7
3804 4140
4 187
482 S
6169
6510
5851
6191
6531
6871
:t41
•J7m
H
7210 7649
78.'8
8227
8565
8903
9241
9,)79
9916
110253
138
■M.i
9
110o90,1109'2G
111203 lll.)99111934J112270
112605
U2940
113275
3609
335
433
130
I!3s;i3;n4^277 114611 U4')44;ll;V278'lli)^ir,
115943
116276 116608
11 694 f)
;;o
1
727 ll 7003 793;
8265 8595 8926
9256
9586
9915
12024.i
330
01
•2
120..74' 120903 12I23I
1215(i0 121888 122216
122544
122871
123103
3525
■',lS
!»;
A
3^iV2 4178 4.)0t
4830
5150 6481
5806
6131
6451'
6781
325
iii)
4
710.. 71^vft 77 .3
8070
8399 8722
9045
9368
9690
130012
323
101
.'.
1.30334 130tJ.>;) 130977113^298
131619 131939
132260
132580! 132900
3219
.!21
IPS
3r.39 3858 4I77 41;)o
0721 7037 7354I 7071
48141 01.33
5451
6709 6086
6403
.(18
■si')
1
7987 1 8303
8618
8934 9219
9561
.il6
\!..b
8
9879 140191
140.;03
140822
141136 141450 141703 142076'142389
142702
:il
i!)U
9
11301.)
3327
3639
3951
42631 4571 4835 6196 5507
5811
ill
:!09
llu
140r23
140 138
14074S 147058
147.3671147670 147985 148294 148603
148911
30
1
9^2 1 9 9rv27
9835150112I50119;1507j6151063;151.370;1o1676;15198.
:(07
(M
o
\ry2i^» l;V2r>94
152900 3205
3510
3815
4120
4421
4728
.503'2
.(05
!)l)
«J
;)330
5040
5943 C2I0
0549
6855
7154
74.57
7759
8061
.'.03
IJO
4
830^2
8004
890) 92()0
9567
9868
160108
160469
160769
16106b
:!ni
1 oO
•
101308
161607
161967
162200
162564:162863
3161
3460
37.58
4055
299
180
43"<3
4Q,jO
4947
5214
5541 6838
61.34
6430
(i726
702^
297
■J 10
7
7317
7013
7908
8203
8197 8792
9086
93 SO
9674
990;
295
210
fi
170202 170.i,)0;i70818 171141
171434 171726 172019:17231 11172603
172895
J93
■270
9
i;.o
3180 3478 3709 4000
4351
4641 4932 6222 6512
680j
.'91
2.S9
170091 170381 Il700ro' 170969 177^i4.9
177536:177.825' 17811 3 178 lul
1786,Si'
•js
1
8977! 92041 9;.V2l !i839;i8012t):i80413:i80690'l809SG 181272'l816.>i
287
c')0
■)
I.'I814!I82129
182415:182700 2985
32701 .3555 3!s39 4123 440.
.'85
81
3
40.1)1 497 J
5259 6542 6S25
61(»8 0391 0674 6956 723f'
.■83
irj
4
7..21' 7803
.808 1 83001 8647
89^18 9209 9490 9771 190051
281
110
;')
iyU33i;lH(l()l:»
190892.191171
191451 !l917.30192010 192289:192507 28 li:
;79
llirf
312.. 340.;
308 1 3>)5;i
•1237 4,.ll 4792! 5.)(;9 5313 5i;2:
.'78
l'.)u
7
.'.900 6170
6453! ('72:>
7005 7!8l 7.V.6' 7832 .SI 07 .';i8.
;T(i
^■24
K
8(i..7 >93'2
92(16: 9!8l
9755 200029 200303 209577 20(W50;2Ol 12 1
.'74
:2^<2
<)
20(39; ;201tihi,J0i:M.i20^2J10/20j488 :;2701 30331 330i) 3u77 3.'.l.
.^72
PI
21
;,
7
10
1 '■'
I.
I).
>\
1 ;',.'
•1
IJ8
.;
\n
C
120
.')
lit;
)t;
I1'2
rK
lOM
21
101
>.s
lOi)
)f
:i!)7
—
...
1.'
.i!i:{
10
:!0()
11
:?:iii
'1
:)i^;!
.17!)
i.T
:i7.i
T)
:!73
•1
:170
■J
36(1
!)
ig:)
—
—
(;
:i(in
1
;!f)7
'J
■■ic>:>
1
■Uri
'
.Ml)
(i
■Mti
■2
,11:5
1
;tu
.T
\:\H
i)
nn
—
— _
f!
t;)3
•■i
tao
ws
1
iiO
^2
ii.l
11
Wl
;!
.ilrf
)
.il6
•.'
:il
i
ill
1
:iOi)
■.
:I07
•:
.iOrt
1
MV.i
b
.Ml
6
I'M)
.J
.'!)7
;
iv.r,
,■.•
l'X\
•J
.'ill
i'
iX9
1
.1^7
;
.'ri.>
(•
.■83
1
.'i
i:
;7rt
7S
I.
J 7 'J
„_
1,00 A urn I MS.
3f53
\'t
7;i
U)
i:i
lo
HI
•JK)
l60iOM20
1 1 6SM
i\ Coir.
3 214188
4
•J04:J!U1201(W3
7()i)6 7365
j)7!<3'ilO(J.)l
21i.l54 2720
4844 610!» 0373
7484 7717 8010
220108 W0370,iiOtJ31,
2710 2:176 3i!38
630!) 5508 6820
7887 8144! 8400
204034
7031]
21031!),
1 2!)8(J
I 6638
8n3J
22()S'J2
34it6
6081
8057
205204
7004
210580
I 3252
1 6902
H530
221153
1 3755
I 0342
8013
•iO>710 200010 20(i'.i80
H441i H710 8'.>7!)
■'11121 2U3S;i211tJ54
' 3783; 404!)j 4311
64301 66!)4 0U.j7
0323! 9585'
'>J I !<;)6 222196
4533! 4792
7115 7372
Pii82 9038
02171269
2119211267
457!>126b
1 722! 261
9810
2224ol)
I 5051
! 7630
230193
262
261
259
268
25G
231724
4264
6789
9299
241795
4277
0745
9198
251038
4064
231079 232234,232488
4517 4770 6023'
7041 7'iP '^'''•'
9550 9800iai0050
242044,242293
4.'j25 4772
0991 7237
<)U3 9087
251H81 '252125,252368
43061 4548 4790
232742I254
62701253
77!>.")1252
24030(1 250
50311242
255514
7918
200310
2088]
6054
7406
9746
271842 272074
4158 4389
64621 6092
255755 2559961250237
8158 8398 8637
260548J260787 261025
>i925 3162 3399
5290 6525 5701
76411 78751 8110
250477 2507181256958 257198;257439 241
8377
261263
3030
69!)6
8344
9980 270213 2704 '.0
272306
4620
6921
2770
5081
7380
3001
5311
7609
;,...,, 93551 9594 9S33
261601261739 261076 262214
3873 4109 434G 4jkS2
0232 6407 6702 6! 37
8o78 8812 9046 92/9
' ' ''271009
3096 3!»27
U002 6232
8296 8525
270679 2709 1 2i27 1 144127 1 377
T^ote&k^021l27^
11281033
2 3301
28126l28148B28"l715l281942;282ir,9
35271 3753
5782 6007
8026 8249
290257 290480
2478 2099
4687 4!t07
6884 7104
90711 9289
4205
0456
8090
4431
0081
8920
290702l290925l291147
2920
6127
7323
9507
280351 280578
239
238
237
235
234
233
232
230
229
280806 228
284<) 3075 227
6107 5332 226
7354 7578 2
93891 9812
'291309 291 591 291 813 292034
"35841 3804 4025 4246
5787 6007 6226 6446
7070 8193 8416 86351219
3001611300378 300595 3008131218
■)
223
222
221
220
^30 301247 301404'30168l301898l302114
302331 1302547 302764l302980l217
20
3412I 30281 3B44! 4059
5566 5781 6996 6211
7710 7924 8137 8351
0843 310056 31026Si310481
2177 2389 2000
4289 4499 4710
6390 6599 6809
84311 80891 8898
311900
4078
6180
8272
320146320354
4275
6425
8504
310693
2812
4920
44f>ll 47061 49211 51301216
6039 6854 700.^1 7282 215
877rtl 8;191 92041 9417 213
3109001311118 311330 311542
3023 3234 3l4r
61301 53101 5551
7018
9106
7227
9314
7436
9522
7646
9730
3656
5700
7854
9!)38 208
322219
4282
0336
8380
330414
243S
4454
6400
8456J
4899
0950
8991
5105
7155
9194
■J30617J3308J9,33_1022 33122
H 84561 8000 Do.'^i ■'"•'
9J340444l340642 340841 i3410a
5516
75031
900 1 1
3310301331832
67211 59261 61311205
7707 7972I 8176204
9805 330008,3302111203
,34i';
2034
30491 33001 4051 4253
5658 5859 6059 OiOO
7659 7858 80581 8257
9050 9349 340047 ;340240
041632341830 20281 2225
2236 202
202
201
200
19f»
193
364
LOCJAHlTfJM*.
pp
N.
1 1 { 3
3
1 *
5
7 1 8
B
n.
•WO
342123 342620 342817
343014 343212
343 109'343606'343802'343999
34419ti
197
19
1
4392 4689i 478.,
4981 6178
6374 6570
6700 6962
6157
19li
3U
a
6363 66 li)j 6744
6939 7135
7330 7525
7720 7915
8110
195
08
3
8306 8500 : 8691
8SS9 9083
0278 9472
9066 0860:350054
101
77
4
360248 350442 360636'360829:351023
351216 351110,351003 351796
1989
1!)3
97
6
21H3
2375: 2668
27011 2961
3147
3339
35321 3724
3910
193
lUi
f.
4108
4301
4493
4685
4876
6068
6200
6452 0643
583 1
192
IX>
1
6020
6217
6408
6599
6790
C981
7172
7303 7554
7711
l!l
lul
H
7935
812.V 8316
8506
8696
8886
9076
9206 0456
9010
l!)ll
171
■230
9836;3B00i:6,a»;0216 360404
360593
300783
360972 361161 361350 301539
189
188
301728'36191736210o 36':29 1
362182
362671
362859 383048 363231 383121
If)
1
3612
3800
3088
4176
4363
4561
4739
4926
5113
5301
188
37
2
6488
6676
6802
6049
6236
0423
6610
6706
ooaa
71011
187
6b
3
7356
7542
7729
7915
8101
8287
8473
8059
8845
9030
18ii
74
4
9216
9401
9587
9772
9958
370143
370328
370513 3706!)8'370883
185
93
6
371068
371253 371437
371622
371806
1991
2175
2300 2544
272.S
181
111
6
2912
3096
3280
3464
3647
3831
4016
4198 4382
4505
181
130
7
4748
4932
6116
6298
6481
6664
6816
6029 6212
639 1
183
MH
8
6577
6759
6942
7124
7306
748')
7670
7852 8031
8210
182
167
9
240
8398
8580
8761
8943
9124
9306
9487
9008 9849 380030
1 ;
181
181
380211
3803921380573
380754
380934
381115
381290'381470'381 650381837
18
1
2017
2197
2377
2557
2737
2917
3097
3277 3466
3030
180
36
2
3815
3995
4174
4353
4533
4712
4891
6070 6249
842.S
179
63
3
6606
5785
60(U
6142
6321
6409
6077
6856 703 1
7212
178
71
4
7390
7568
7746
7923
8101
8279
8150
8634 8811
8;)8!i
178
89
5
9166
9343
9520
9098
9875
390051
390228 390405;3!)0582'390759
177
IOC
6
300935 391 112'3912B8
391464
391641
1817
1993
2169
23 15
2)21
170
1'21
7
2697 2873
3048
3224
3400
3575
3751
3!)26
4101
4277
170
14 J
8
4452 4627
4802
4977
6152
632()
6501
6076
6850
602,.
175
159
9
6199 6374
6548
6722
6890 7U71
7245
7419
7592
77liO
174
173
250
397910 398114
398287
398461
398034 '398808
398981 1399151
399328 309501
17
1
9674 9817
400020
400192 400305400538; 40071 1
400883 101056 401228
173
34
2
401401 401573
1745
1917
20^9 2261
2433
2005
2777
294!)
172
51
3
3121
3292
3464
3635
3807 3973
4149
4320
4492
4<U;3
171
68
4
4834
5005
6176
6346
6517 6088
6858
6(J2<)
6199 6370
171
85
6
6540
6710
6381
7051
72:; 1 7391
7501
7731
7901 8070
170
102
6
8240
8410
8579
8749
8918 9087
9257
9420
!)5!)5 97l'>4
lOJ
119
7
9P33 410102
410271 4I0 110
11000!) 410777 110940
11I1U4112M3'41145:
10!)
136
8
411020
1788
1950
2124
2293
2461
2029
2790
2904 3iai
168
153
9
3300
3467
31)35
3803
3970
4137
4305
4472
4639 4800
167
167
260
414973
415140
415307
415474
415041
415308'415974
416141
4I6303I4I6I74
16
1
6041
680T
6973
7139
730(5 74721 7638
7801
7970 8135
100
33
2
8301
8K)7
8033
8798
SriOl 9129
9295
9400
9025 9791
105
40
3
9956 420121
420286
420451
420010420781
420945
121110
121275 421139
165
60
4
421604
1768
1!)33
2097
2201
2126
2590
2754
2918
308:
104
82
5
3246
3410
3574
3737
3:)01
4065
4228
43!) i
4555
4718
101
98
6
4S82
6045
6208
6371
6534
60!)7
5S0O
6023 6186
6349
103
llo
7
6511
6074
6830
6999
7101
7324
7486
7018
7811
7!)73
162
131
8
8135
8297
8459
8621
8783 8944
9106
9268
9429 95!) 1
162
148
9
270
9702
9314
430075 430236
430398 430559 430720'430881
1 __ 1
431042431203
1
161
101
431364
431525
431 085 431 846 43v;o07 ^ 432 1 67 i 132028.132488
432G49I132H0!'
IC
1
2;»t>i>
3130
3290 3450
3010
3770
3930
4090
4249 440!;
100
3J
2
456!)
4729
488;i 5048
5207
5307
6520
5085
581 1 600 1
1 59
47
3
6103
6322
C181
6040
6799
6:)57
7110
7275
7433 V592
159
63
4
7751
7909
8007
8220
8384
8512
8701
8859
9017 9175
158
79
c
9333
9491
9648 9806
9;)64
440122 440279
440437
440594'440752
158
90
6
440909
441006
441224 441381
4''.1538
10i)5
1852
2009
2106
2323
157
111
I*
2480
2637
2793
2950
3100
3203
3419
3570
3732
8889
157
126
8
4045
4201
4357
4513
4009
4825
4981
6137
5293
6449
156
142
9
6604
6700
6915
6071
6226
6382
6537
6692
6848
7003
156
1.
n.
■m^
._
>t;
l!)7
II
lOli
10
!!).'>
31
191
nil
1!):)
iii
iq;i
ii
i(ij
It
nil
ifi
MM)
i!l
IM9
21
\M
II
ISH
ill
187
10
18ii
:(.'!
I8..
:.s
181
J.i
181
»1
I8:»
k;
IHi
JO
IHl
—
i:
181
}(i
180
:.s
179
•i
178
¥.t
178
'■>•>
177
:\
17(>
17()
1 ■?'»
JO
174
)i
17;i
is
17:!
1!)
I7J
i;i
171
•0
171
•0
170
u
l{)';i
ii
\n<)
i2
168
10
107
—
"1
107
!.,
Kii!
M
10;)
Ui
lOf)
iC
104
b
101
10
103
■;j
16'J
)i
16>
in
101
)<•
!0I
);:
100
51
1 ,■.!!
)•:
1 ■')!>
(.J
158
V.'
158
in
1.1?
=i<t
157
w
156
33
155
L0(SARIT1IM8.
r>n5
r
PI'
N.
280
; I
ti
3
4 5 1 6
1 '
: 1182 12
«
°
1).
156
117158 4473ia!4474U8
4470231 14; .78
4479!13 418088
448;i97!4'.'J552
lA
1
8700 8801 1 UOl.)
9170
I);t24
9478
9H:I3 9787 09ir4ftO095
154
;»!
2
J.")024!) 450403
450557
450711
450805
451018
46)172 451326 451479
1033
154
40
3
1780, 1910
209:1
2247
2400
2553
2700
28591 3012
3165
tu3
(il
4
3318 3471
3624
3777
3!t30
4982
4235
4387
4540
4092
15;i
77
6
48451 4:)!)7
51.;()
6302
5154
6000
6758
6910
0062
O.'l 1
15J
nv>
0300, 0518
6070
6821
097 :»
7125
7270
7128
7579
7731
152
107
7
78821 8033
8184
83;i0
84>m7
80;ts
8789
8910
9091
9242
151
lii
8
93921 9543
9694
9815
99!»5;460I40' 160290
400447 460697 ;40O74t.
151
i;)8
9
290
100898401018 401198
461348'U3149y 1049
1799
1948 2098 224'5
150
150
102398^402548 462097
462817! 402997
4fi3146
403290
463145l4«3594^463711
If)
1
3893
4042
4191
4;) 10
4190
4039
4788
49;i6
60851 623 1
149
a!»
2
5383
6532
5680
68.!9
5977
6120
0274
0423
6571
6719
149
41
3
0808
7016
7104
7312
7100
7008
7750
7904
8052
82J0
148
SO
4
8347
8495
8013
8790
89:18
9085
9i;!3
9380
9527
9675
148
71
6
9822
9909
4701 10
470263; 170110
470557
470704
470851
4709981471145
147
88
471292;471438
1585
1732
1878
2025
2171
2318
24641 2610
140
io;i
7
2750 2903
3019
3195
3:) 11
3487
3633
8779
3925
4071
140
118
8
4210 4302
4508
4053
4799
4914
5090
6235
5381
6520
140
13.'
9
300
5671 6810
5002
6107
0252
6:)97
6542
6687
0832
6976
145
145
177121 J477200
477411
477555I4777OO
477844
477989
478133
478278
478422
14
1
8560
8711
8855
8999
9143
9287
9431
9576
9719 9863
144
w
2
180007
480151
480294
480138
480582
4S0726;480H09
481012
481166 481299
114
43
3
1413
1580
1729
1872
2010
2169
2302
2445
2588
2731
143
67
4
2874
3010
3159
3302
3 145
3587
3730
3872
4016
4157
143
72
5
4300
4442
4585
4727
4809
6011
5153
6295
5437
6579
142
80
5721
5803
6005
6147
6289
6130
0572
6714
6855
6997
142
100
7
7138
7280
7421
750;i
7704
7 815
7980
8127
8209
8410
141
114
b
8551
8092
8833
8974
9114
9255
9390
9537
9077
9818
141
l2'J
i!
310
995S
490099
490239! 1903a0 190520
190061
490801
490941
49108l491222
140
140
19 1302 [49 1502
491012;491782U91922
492062 492201
492341
4924811492621
14
1
2700
2900
30 10
3179 ;}319
3168
3597
3737
38761 4015
139
28
2
4155
4294
44:13
4572 4711
4850
4989
5128
6207 5100
l;39
41
3
5541
5083
5822
5960 6099
6238
6370
0515
6053 6791
1:J9
55
4
0930
7()(i8
7200
7314 7483
7021
7759
7897
80:t5 8173
138
6!)
5
8311
8418 8580
8724 8862
8999
9137
9275
9412 9550
138
83
C
9UU7
9824' 9!»02 500099 500J30
500371
500611 inOOO^b
500785 500922
1:17
97
t
^01059
501190,5013;!:;
1J70
1O07
1741
1880
2017
2154
2291
137
no
8
24i7
2504 2700
2>t:J7
2;)73
3109
3246
3382
3518
3055
i;^.o
124
320
3791
39271 4003
1
4199 4335
4471
4007
4743
4878
6014
130
130
505150 50528050642l 505557i505093
505828
5059641500099
506234 506370
13
1
0505
6010
6776
0911
7(140
7181
7316
7451
75801 7721
135
27
o
7850
7991
8120
8260
8395
8530
6004
8799
8934! 9008
135
40
3
9203
9337
9471
9006
9740
9874
510009
510143
610277
510411
134
54
4
510545
510679
510813
510947
511081
511215
1349
1482
1616
1750
134
67
C
1883
2017
2151
2284
2418
2551
2684
2818
2951
3081
i;i3
80
3218
3361
3484
3617
3750
3883
4010
4149
4282
4415
I3:i
94
7
4548
4081
4813
4946
6079
5211
5344
5476
6609
5741
133
107
8
6874
6006
6139
6271
6403
6535
0668
0300
6932
7061
132
121
9
7190
7323
7400
7592
7724
7855
7987
8119
8261
S382
132
330
518514
518640 518777
518909
519040
519171
519303
5194:14
519560519097
131
13
1
9828
9959
520090
520221
520;! .3
520484 520615
520745
520870
521007
131
20
ti
:)21138
521209
M:lO
1530
1001
1792
J 922
2053
2183
2314
131
30
3
2114
2575
270rx
2835
2i)00
3090
3220
3356
3180
3016
130
52
4
3710
3870
4000
4l:J6
4206
4390
4526
4656
4785
4915
130
6,>
I,
5015
5174
o:;04
5434
6563
6693
6822
5961
6081
6210
129
78
0339
6469
6598
6727
0350
0985
7114
7243
7372
7501
129
91
7
7030
7759
7888
8016
8115
8274
8402
8631
eooo
8788
129
104
8
8917
9046
9174
9302
9430
9559
9687
9815
0943
530072
128
117
9
530200
530328 530456
530584
630712
530340 530968[531096!531223
1351
128
3!;»;
l.tXiAUl ril.MS.
I —
I' ;
Uj
1
•J I
M
4'J
til
T:i
t<
•J
110
1
•il
art
lii:
71
(1..
ID?
Ill)
I
l.'jd o 1 l(t«H
&:i()7
(r)4.i
:i 777i')
ll 90(»;t
()
7
i)
I
•i
a
38h;)
6;)I7;M
3t)0!»
4iM
6>47
(iHll
8071
mii
fl4()i.bO
3074
.3IMW 03i!>n(ii):i:in
4407
r>n;4
0!i;)7
Hn»7
04.V.'
0I0700
l!to3
31119
M41M64431(}'.">4444()
oiai
Htjtiti
78»«
•1789
80'Jl
9li« 9il9
r,6«:i;) I 660473
3:i;i
4):ii
6HO0
7003
1(:>78
sioasodjoitsf)
ao7H aj03
33i3 3447
3ti4."
3:191
3&1S
4tit!l
47H7
cmi
00o3
7 189
7315
8448
8574
9703
98'i9
4914
tilHO
7441
8009
377 J
AdII
Haoo
7.167
8Hjr>
541080'>41'i0d
'J3i7
3671
99.J4;a4«079
1130
a»J9« 38J0
63'it>'.'7
3( ■<".)
M(I7
ti lai
7ti93
8961
I140J01
14r»l
•i701
3914
K)7i
a790
4(K)4
a'J16
l(i94
'J9I I
41'i(!
6330
i06303
7607
8709
9907
,>01101
•J^'1I3
348 1
4600
6848
7l)J0
6664J3J
7027
88i9,
6600'2G
vni
'J41'i
3600
4784
6966
7144
666644
7748
8948
660146
1340
'J.')31
3718
4903
6084
726J
6678
6913
8144
937 1
66o;iy.)
1816
3033
4i47
6467
644.>64,fl4468H
680i 69J6
7036 7169
HJ67 8389
91!)l! 9616
660717 660810
1938
3166
4368
6678
•J060
3J76
4489
6691)
6448 1i
6049
'■iSl
8612
9739
550962
!218ll
3398
4610
6820
666664
7868
9068
560265
146(t
2660
3«37
6021
6202
7379
66^78;")
7988
9188
,560386
1678J
2769
3966i
5139
0320
7497
666906
M108
9308
660604
1698
2887
4074
6267
6437
7614
544916
6172
7J06
8636
9861
661084
2303
3519
4731
6940
146060
829ti
7629
8768
0984
551206
24'26
3610
48.V2
0061
l2h
127
127
126
126
120
125
126
126
124
667026!
8228
9428
66062 1
1817
3006
4192
6376
0656
7732
646 1 83
6lt!)
7062
H88I
650106
1328
2647
37<i2
4973
6132
: 657 146 567207
8349 B469
6686711608788
98421 9969
67101067n26
2174 2291
33361
4494
60H)
6802
7951
9648
660743
1936
3126
4311
6494
6073
7849
660863 O80982
0667
067387
8689
9787
121
124
123
123
123
122
122
121
121
\1\
2065
324't
4120
5612
6791
7967
2171
336 .
46 IS
6(311
6909
120
120
120
119
119
119
9
118
118
808 1 1 18
9097
3462
4610
6766
0917
8066
0212
,,)6b»06:
'670076
1243
2407
3668
4720
6880
7032
8181
9326
669023;669140'609267
670193 5703lt9;6T0l^()
II
22
3
41
66
06
77
i?S
9i
390
1
J801;6
1267
2404
3.i39
4070
67 99
0926
8047
9167
590284
6S0241
1381
2618
3652
4783
0912
7037
8160
9279
1369
2623
3084
4841
5906
7147
8296
9141
691066
2177
3286
4393
64r.l)
6697 j
7696'
8791
9383
600973
691176
691287
2288
2399
3397
3608
46(13
4614
6606
6717
6707
6817
7806
7914
8900
9009
91)!J2
Gonioi
601082
llDl
680356
1495
2631
3766
4S96
6024
7149
8272
9391
590396 590607
591300
2610
3618
4724
6827
6;^'7
S024
9119
000211'.
1299
591610
2621
3729
4834
6937
703
8134
9228
680469
1608
2746
3379
6009
6137
7262
8384
9603
500019
1476
2639
3800
4957
6111
7262
8410
9666
16!».
2
3916
507
6226
7377
8626
9669
117
U7
117
110
116
116
116
115
116
114
680583
1722
2368
3992
6122
6260
7374
8496
9616
590730
691621
2732
3340
4946
C047
7M6
8243
0337
600319,000428
1408 161
680697
18361
2972
4106
3235
6362
7486
8608
9726
690342
691732
2843
3960
6066
6167
7266
8363
9446
600637
1626
591843
2964
4061
6106
6267
7366
8462
96:6
600646
1734
601966592066
680811
I 1960
I 3086
4218
6348
0476
7699
8720
9838
590953
114
114
114
113
113
113
112
112
112
112
30C4
4171
6276
6377
7476
8572
0665
600765
laiaj
317;:
4282
6386
6487
7686
8681
077
600864
1961
111
111
111
110
no
no
no
109
109
;09
Uj
M((7
t) VM
7til>:J
ti(ir)i
a40'.'0l
1151
•J70I
3!) 11
.t0 60Wi<7
i9r)o;6920{!f)
(064
1171
V27U
j.'J77
7170
W72
[)60r)
9765
317;:
4Jb2
63ri6
C487
7r)S0
86S1
977=1
600864
1943 19ol
l,0{J.v.:; TIIMS.
art?
• N.I j 1
9 3 i:a{<i7 fi{B
I).
UHKiOjimomi'jKii
!rtii'W7;nO'/3H») tioji'M (jo;i)ii;i iio.'7ii (JojiiiiitiO'jni vii(i3030
lOis
1
1 .'Jill li::,:
330IJ 310;» 3.7V iJ'iMU 3791 it'll'.'
401(1 .11 M
lim
•.'
•j' 4.',ii 4:i:j
ail'^l 4.>')0 40.)S; 47mii 4i'i74 49.s.i
7>(»''9 ft 197
IDS
.1
4" Oifii: Ula!
r.yiii nam tyV.u;. muI mc,i\ m.M
tilli 6274
10%
I:
(iV'ili (!7(f4 (lilli «9I!I Tthfii 7133
72411 73 ^^
107
h
fl 7l.,V 7.)»j_
7«09 7:77, 1'iSi: 7901 H(t98 HJ(i,,
83121 8119
107
ti
H.,'.'(l Htt;),)
87101 est: w'i.,4: notii' oio*' 9.'7il 9;mi! im'Im
107
7.
7 0.i!»l DTOI
9808! 9911 OHM, Jl lilOl J.h 010J34 HIO:! 11 i(illlll7 lUO.>.yi
107
m
H illOti'iti tll07(i;
!010.i;3 0IO!l79, lOiti incj, 1j;h, 140,)
1611 1017
100
l^
9. IVJ.
tlOOI'^Tri
1 l.'v.',
1 l;*3(i •JOli' all.S iJi.il 'I.IW
•J 100
01362*
2.»72 *207m
100
106
lOUSilO
Oi:99()()13IOJU13.'0;OI3;.l3 01311i)
013030 01373(1
II
1
;i!!j ;4;vj7
•10;,;
1 41. 19 4J(il 4370, 417.)
4.VII
40.S0! 4792
100
Vl
4
4 so 7 Olio:!
/.IOC
i 6J13 63191 f)144i 6<.,'9
ft«3t
rnu
aHi6
10.)
;).
'
WI.iMl fi(l.>.1
(illli
1 6.;ej.v tf;i7l
017U! fi.)dl
0080
679(
♦))96
106
I.
^
7ooii; 7io.>
I'iH}
731.)j 74 io! 7.i"i.">j 70.'9
7731
7831J
7913
1 06
Xi
/
t.OlH, ttA:i
Sir,:
8.10. •' 8400! 8r,T\\ ^,\m
8780
8SH1
8989
106
u:
(
mi'.fl' 919,i' Ji;iO.i^ 9l(Mi 0/1 ll »'tll.j; !i;io
1182!
992.s'!i2tl(i3'J
104
1 i
7
t;2l)K;i; WOMO 0.'0:)4 1 (iJOl H (WO.,.,^ 0i')9.)(l 0'J07:Mt U'JOH*; 1
020908 1072
104
> I
s
117ii l'.',<o :«si' Mis. l,,9i' KiWi 170!> luo:!
201)7 2110
,...
!)„
1J(
ini^ :j;iiM
1 awi
'(}.>34..<
' i.n.fi Mif
1 273'.
1
o2370»
!l 2.'^36l 2939
.SO 13 3110
lui
103
Gg:^•.•liMi•i.^;^,■,;^
tii.?.^)')9 02300.1
i 023809 020973
021070^02417!)
If:
1
4'.'S.
4;H.V 44.Ss: 4.')9!, 4t)0r> 470H! .1901; .'.01)1
61l)V 6210
103
•ii'
k'
.Mlv{ ;)41.. (V)l.^
60.M 67'Mi M27l 6920, O.I.J'
0136 023M
103
;ji
;i
(i:i 10 1)4 1,! ().■) 1(
0«t8; 07.).H 03)31 (i9.i(i! 70.)S
7101
72S3
103
11
4
I'Mii 74!i,i 7.,7i
7ii73i 777i^ 787h, 79.Siii WisJ
818.
82 A 7
102
.li
.';
tH.i'.ij Hl!)i H')9;i
8rt0.) 87n7 89001 9002' 9101
9201
1 9308
102
01
SUKil i;,lj! !((il:i
1 971.) 9H)7; 9919 03002 1,030 I23l030.i.'ri;.30320
U)2
71
/
oaoiv.soaooiio'ti.'joiWi o;jo7;i:i (Woh;!.. oauo.yj i.y.hu luv
1241 1312
102
H.'
tr
ii4i j.M, Kiiri j74s; i.!i!) i'.).,ii io.vj: 2ii<;:
2266 2360
101
!),•
•Jl);i ao.)9 •2(Um\ i>7tilj UfSiij
1 29(>.'
'033!)73
1 .'1001 310.:
.•;.'Oii 3307
101
101
i;i;; IO.S'(i;!:j:,o:) 033'!70!(;3:?77l;»i:):)i(j
03 1074.034 1 :6;03 1270 034370
10
1
417:
4;.7.S 4079; 47701 4.iS:' 4:)H1
j iM);s)i 61821 62.S3
6383
101
■n<
'
;Vt:JI
6.iSI fi(i;.:.)
.578''>i ()SM
.)9ti0
B087I ei87 6287
0388
100
;)'
u
(ilSi
«•).■<•! eu.io
ti78i
fW.i9
C9,i:) 7(;mJ; 7l,Si 7.,.;i(
7;;3o
100
4i.
•1
7400
7u:)0 "G.'iO
7;90
7890
7990 S090I (1190 8290
S389
100
,')(;
'■
H4S'.)
Hib'.t eo.io
87. iO; l^.i'iS
8;'8:.i 00081 9 If..; P2.S7
93^7
too
(•!•
li
!>tS(i' OOHtij 9(M(;
97.S")! 9iti.. 99.S4;04O0.:4 040l.S3'.Jl02:'3'0103S2
99
/'I
7
ol'J4f>l,( O.JSl
t)40ti80
010779 (;!0il79:04097.i; 1077; ll7? 1270' l.7r>
99
'•.'I.
h
li71
lo73
107.I
1771 lS7i 19701 20091 2108 22071 2360
09
'M
lid
•J4;h..i •J:,o:i
20'J>
2701 2duJ
2969 j •mm] 3161)
1)43;) iO 044011044143
3266 3364
99
98
i:j l,,3
(U3,i.=v 1 \i43t>.30:i) 137 V.i'm I3J 1/
0S4212'ti4434O
U)
1
44:{j
4.);: 7
4l)3rt
473!
4 1321 4i''31
6029] 6127
6220 1 6324
98
'^'<l
64.','
(■<;):.' 1
601!'
6717
631.^1 6913
eon; Olio
O2OC1 030()
98
:ili
•'
(iJOt
0.0.'
(iUOO
Oj!)y
679'.:
089 1
0'V)i; 708!)
7 1ST
7286
98
o.>
'1
7;Ji;i
7l.il
7o79
7070
777 1
7;i72
7909 1 8007 8106
8262
9S
4;i
■■
y;;'iO] tii'i.s
8,0.i6
80.)3 87."i(i
8.:i48
8946 9043 9140
9237
97
.>;<
(;
9,i:).i 9i;i'
Oj.IO
9027 9721! 9o2l
99i!;(i6091O06O113''J6O210l
97
d *
7
K'iKOS tioOlO.) !(ji')0.")0'.'
()6O.")990J0r;!>:i!0)O793 35flS90j 0987 lOSlI 1181
9/
7t
b
I:J76J 1.170 147^'
1609 lOOOl J702i I;^.i9 1.960 20131 21.i0
97
bi
o
!i::.'4tv jai.'! '.'440
2.i30 2033 270,)
2.SvO'
1
0.:.3791 i
2923 j .1019 3110
97
PO
lot.
i ).')•.'! 3 6033 Jo'f)j:nOj]
i63602i063.:96'0J3096
;63v;8.'<'0639:^4'064080
10
J
4177i 4i7.'M
43001
44 J.. 4.,o:;
4i;.Vj 47.it; 4.!.M)i 49101
6042
O.i
l!)
'^
ol:i.il
1
j"i3.i
633! 1
.')42:
!>:;■••■)
AHV.)
6716' 6 J 10
fif.'OO
0002
90
•Ji;
«i
OCCLJ'
liiDl
CJ90
03a0
Ol.vi
6677
0073 67tJ9
0801
ti;'60
90
3S
4
70.")0
71.V,!
VJ17
7313
74 3
7534
7029 772.1
7.020
7910
90
4ri
.'■'
you
8107
SJOi
b29.j
83.J3
81Sd
8684 8079
8771
fe;i70
96
Oti
1;
HOo ">
9000
91;V>
92)0
9310
9i.!)
9630 9031
9720
9821
96
07
t
!i:il« (JGOOU li'JOlOtiOtiOviOl 'OCJlW.itli'aSOoOii
000480 000.)81 '000G7O00O;7 li
95
77
e
iOOHliOi Oi.'liOJ 10).")
1160 121.J 1339
1434,' 1629 1023
1716 96
Sti
<j
1813 1907 200^!
2090 2191 2280
23801 247.^ 2609
2063 IV.
308
LOGARITHMS.
. ^'r
I'P
N.
*30
! 1 2
3
4
5
6
7
8
' 1
a.
94
31V27.38 (it>28'>J 0(W9 47
t)63041
363135 683a306fi332.l'
353418 66351 2!668607
9
1
37011 3?90
3889
39.s;i
40781 41721
4266
4360
41.;4
451S
94
1!>
()
4*i4i! 473*)
4830
4924
5018
6112
5206
5299
5393
6187
94
■J8
;;
6.)8l 6ii7r.
5769
536;
59..6
6;)50
6143
6237
63:j 1
6.124
94
:!8
,1
rt..l::i Ooie
6705
6799
6392
6983
7079
7173
•; 266
736('
94
17
1.
74.j3 7ii4ti
7640
7733
7826
7920
6013
8106
8199 8293
93
",6
f
838C.; ^470
8572
8065
8759
y3rji
8945
9038
9131 9224
93
C,6
7
9ni7 !)1I0
9503
9590
9689
Otiii
9875
9967
370060 67015S
93
76
8
H70y4(i;tJ7033y f;70431
670524 6706171
f)?0/10
070802
670395
0988
1080
93
U6
470
11731 \26r, 13,^8
1451
1543
1636
1723
1821
1913
•2005
93
92
a720!/ci,(i7i!)90
672283
672375
672467
072560
672652
fi7'2744
672836
672929
9
I
3021
3113
• 3205
3297
3390
3482
3574
3666
3753
3850
92
18
O
3!»40
4034
4126
4218
4310
4402
4494
4586
4ti77
4769
92
•28
3
43(Jl
4953
50.15
6137
5228
5320
6412
6603
6595
5687
92
:)7
4
6778
5870
5962
C953 6145
6236
6228
6419
651 1
6002
92
46
6
6Ui)4
6785
0876
6968
7069
7151
7242
7333
7424
7516
91
.)6
6
70'07
76<'H
7789
7881
7972
8063
8154
8215
8336
8427
91
i «J4
7
Sf>\S\ 8B0it
8700
8791
88.82
8973
9064
9155
9246
9337
91
74
,S
1)4 281 r.)19
9610
9700
9791
9r)82
9973
680063 6801541
630245
91
Hi
9
i
dK133tiiijS0420
630517
530607
680698
680789
080879
0970
1060
1151
91
kV. ■si.^.]iin3l3:t'>i
631122
681513
681603
681693
681764
081874
681964
682055
90
9
1
214,; 2^3.)i
232<i
2416
2506
2596
2886
2777
2867
2957
90
13
2
3017
3137
3227
3317
"407
3497
3587
3677
3767
3857
90
'J 7
3
3;'47
4031
4127
4217
4307
4396
4186
4576
4666
4756
90
36
4
484:.
4:';;.;
502,,
51'4
5204
5294
5383
5473
5563
6652
90
4.)
^
5; 4 2
£831
5921
6010
6100
' . .9
6279
6368
6458
6547
89
54
6
61>:)rt
6726
6815
6901
6994
. j83
7172
7261
7351
7440
89
63
7
7i2ii
7618
7707
7796
7886
7975
8064
61.53
8242
a33i
09
1=1
8
81.0
8i09
8598
8687
8776
8865
8963
9042
9131
9220
89
81
U
9301*
93;)S
9480
9575
96C4
9753
9841
9930
690019690107
89
89
490
Gnoi9e'ti:w23)
690373
690462
690550
690639
6907'^S 16908 16
690905
690993
9 1
;'>si
llTO 1258
1347
1.435
1524
1612
17i)0
1789
1877
88
a 2
1960
20.; 3 1 2142
2230
2318
2406
2491
2583
267 1
•2759
,S8
2.)
3
2S17
2035
30; 3
3111
3i:i9
3287
3375
3463
3551
363:»
88
S)
4
3727
3810
3903
3991
407 .s
4166
4254
4342
4130
4517
88
41
(i
4t)05
4693
.■1781
4868
495'i
6044
6l'U
52 1 9
6307
5391
SS
d:i
6
5482
5.;6.»
5(i.J7
5744
6832
5919
6007
«K)9i
6182
626! t
87
6
7
0356
6441
or.:, 1
6618
6706
6793
6880
b'968
7055
7112
81
7(.:
8
7229
7317
7401
749 757.8
7 665
7752
7839
7926
8014
/i.'
1)
500
8101
818;i
8275
8302 8149
G.j35
8622
8709
8796
8883
87
87
698970
699057:699141
(ir9231 '6993171699104 699.191
'699578
69966 ll699751
!)
i
9838
9924!70(t01l
700098:700I84i700271i7003.)8
700444
700531
700C17
87
17
2
700704
700790! 0877
0963
1050
1136 1222
1309
139.^
1482
86
'It;
3
1568
16541 174!
182/
1913
1 999 2086
2172
2258
2314
6(i
3!
4
2431
2517 2603
2689
2775
2861
2917
3033
,3119
320;
60
4.!
5
3291
33771 avs:
3519
3i;35
3721
3307
3893
397!
406;
86
oZ
G
4151
42301 482.'
4408
4494
4.)70
4665
4751
4837
4922
86
ec
7
6008
6094
sr.^s
C265
6351.
6436
6522
6607
6693
677t
86
ti!)
fc
6864
6949
eo3i
6120
6206
6291
6370
6462
6647
6632
85
77
1
510
6718
6803
C88o
6974
7051
7144 7229
7315
740(
743:
85
707o70707655 70774U
70782 3 707911
707996 708081 ;70816ti
► 708'?5l
70S33f
85
i.
1
81211 85061 85!) 1
8676 j 8761
8846 89311 9015
9101
918:
85
n
i
9270
fj')r,n! t^^ »(
95.; i. 9699
9(<9l <)77!>l 986.*!
!1«UH;71(i03r
85
r
S
710117
710202!710287
710371
710156
7IO0II
710625 71 07K
71079
I o.^7r
65
3
4
0963
104!:
113.
1217
1301
138;
147r
155
163r
172?
84
4.
£
1807
1893
197C
206C
214
222r
» 231?
2397
2481
25(;(
81
5f
e
26 DO
2734
I aei.
29Uv
, 298b
307C
» 31.;5
I 323f
) 332:
( 340"
84
oiK 7
3491
357;:
3651
374.
382e
39 U
) 3991
l 407 f
! 416
! 42461 84
(i/'l (
433(1
441<l
[ 4197
4581
466:
474rl 483;
1 49ie
i 6001
) 508t 84
7>
i ^
5167
6251
633.^
o41i:
i 650.
658.
> t'M
) 6?5C
! 683(
; 6B2( ^
t 84
3
3
4
6
6^
■21668607
4.V1H
6!S7
6121
TMv
829!i
9.>24
6701oS
1080
2005
•>
1
i6l
ii
3
ts
i8
7
»o
1
>4
!()
16
i4
50
>4
57
57
56
53
38
■)l
1i
M
19'690107
672929
3Bi)(i
476'.)
5687
6002
7016
8427
!):i;J7
63024 ■>
1151
682055
2957
3857
4756
6662
6547
7440
a33i
9220
a.
94
94
91
94
94
93
93
93
93
93
92
92
92
92
92
91
91
91
9i
91
69099;,!
1877
27 i9
3639
4517
5394
6269
7112
8014
S8J3
699751
OOCl
14S2
2314
320;;
406;
4922
5778
6632
748;
70S33fi
9185
UH 710033
f.4
0^79
65
39
1723
84
181
2o6i)
81
153
3407
91
02
4246
84
100
5084
84
J3G
W)20
84
00
90
90
90
90
89
89
09
89
89
89
88
H8
88
8S
SS
87
y,
8i
87
87
87
86
8(i
66
86
80
86
85
85
LOGARITHMS.
369
370
LOGARITHMS.
ii 1
pp
N.
'
3
3
4
5
6
7
8 I 9
0.
58i'
76:)l'.'8'763503!763.y78
7S3053
763727
763802
763877
763952
764027 764101
75
7
1
4i;6i 4iol 43.!«
4ii>.)
4475
4550 4624
4699
4774J 4813
75
1.)
•^
4y;!3! 41)98
5072
5147
6221
5296 6370
5445
5520
55!)4
75
•:■:
3
566:): 6713
58! 8
6892
5066
60 41
6115
6190
6:64
63:i8
74
30
4
6413 6l'i7
6562
6636
6710
6785
6859
6933
7007
7082
74
37
5
7156
7230
730 4
7379
71.3
7527
760 i
7675
7749
7823
74
44
6
78!tS
7972
8046
8120
8191
8263
8342
6416
6490
8564
74
iVi
7
863S
6712
6780
8860
893 1
9008
9082
9156
9230
9303
74
5!)
8
!)37
9151
9525
9599
9073
9746
9820
9894
9968i7700al
74
07
590
770115
7701 6\l 770263
1
770336
770410
770W1
770557
771293
770631 77070 '>i 0778J
74
7708oi
770926 770999 771073
771146
771220'
771367
771411)177 1514]
7)
7
1
1537
1661
1734
1808
1881
19)5
2028
2102
2175
2248
73
U.
'2
V>3>.'
2395
216t
2642
2615
2688
2762
2835
2908
2981
73
•>o
;i
3055
3128
3201
3274
33 18
3421
3494
3567
3610
3713
73
■2\)
4
37o6
SSuO
3933
4006
4079
4152
4225
4298
437 1
44W
73
37
5
4517
4590
'Hvli
4736
4809
43S2
495.,
6023
51(10
6173
73
44
6
6216
63!!)
5392
5465
5533
5610
5683
6756
5829
5902
73
ul
7
5974
604 7
6120
6193
62i)5
6338
6411
64.^3
6556
6629
73
y«
vS
6701
6774
6840
6919
li'Ml
7061
7137
7209
7282
7354
73
6ti
SOU
7127
■ 7499
7572
7614
7717
77S9
78S2
7934 8006
8079
72
72
(78151
778224
778296
778368
77CSU)
778513
77358;,
778658
7787;?0 778302
V
1
8874
8917
9019
909 1
9163
^ 9236
9308
9380
9452 9524
72
14
•2
95i)6
906!)
9711
9813
9885
9957
780029
780101
780173!780245
72
•2'i
3
780317
780389
7804(il
780533
730605
780677
07l!i
0821
0893
0965
72
Hi)
4
1037
1109
1181
1253
lt24
1396
146s
15 40
1612
1 634
72
3(i
5
1755
1827
1S99
1971
2012
2114
2186
2258
2329
i401
72
43
6
2473
2544
2616
2688
2759
2')31
2902
2974
3016
3117
72
uO
<
3180
3260
3332
3403
3475
3546
3618
3689
3761
3332
71
6rf
M
3004
3975
4046
4118
4189
4261
4332
4403
4475
4546
71
6,)
9
4617
4680
4760
4831
4902
4f)74
501.',
6116
6187
6259
71
filO
735330
785401
785472
785543
785615
785686 785757
785828 785899
7S5970
71
7
1
6011
6112
6183
6254
6325
6396
6467
6538 0609
3680
71
14
.1
6751
6822
6893
6964
7035
7106
7177
7218 7319
7390
71
'iJ
3
7460
7531
7602
7673
774 i
7815
78S.1
7956 8027
8098
71
■2H
4
8168
8239
6310
8381
8451
8522
859;;
8663 8734
8804
71
36
5
8875
8946
9016
9087
9157
9228
9299
936:; 9410
9510
71
43
6
9581
9651
9722
9792
9863
9933
790001
790074:790144
790215
70
f)0
7
790285 790356
790426
790496
790567
790637
0707
0778
0848
0913
70
57
H
09S8 1059
1129
1199
1269 1340
1410
1480
1550
1620
70
64
1691 1761
1831
1901
1971
2041
2111
2181
2252
2322
70
792392:792102
792.J32
792602
792672
792742
792812
792882 792952
793022
70
1
1
3092
3162
3231
3301
3371
3111
351 1
3581
3651
3721
70
14
*
3790
3860
3930
4000
4070
4139
4209
4279
4349
4418
70
:il
3
44;"
4558
4627
4097
4767
4836
490(;
4976
6045
6115
70
28
4
6185
5254
5324
53i)3
6463
6532
6602
56721 6741
5811
70
30
5
5880
6949
6019
6088
6I0S
6227
6297
6366 6 136
6505
69
4J
6
6574
0641
6713
6782
6S52
6M21
6ii90
7060 7129
7193
69
40
7
7268
7337
7406
7475
7545
7614
7683
7752 7821
7890
69
56
8
7960
8021)
809o
8167
8236
8305
8374
8443! 8513
8532
69
63
!)
630
8651
8720
8789
8808
8927
31)96
S06,
9134 9203
9272
09
799341 1799409
799478
799547
799616
799685799754
799823 799892'799961
69
7
1
.■!00029 300098; 800 167
800236
300305
300373300H2
80051 18005801800648
69
14
•2
0717
078;;! 0«54
0923
0992
1061
1129
11981 1266
1335
69
21
3
1404
1472i lo4i
I6O1)
1678
1747
Islo
I8is4 49..2
'M2\
69
ZS
4
2089
21581 2226
2295
2363
2432
2500
2568 2637
2705
69
68
■do
2774
2842
2910
2979
3047
3116
3184
32521 3321
3389
41
31571 3525
3594
3662
3730
3798
3807
3935I 4003
4071
63
4a
7
4139
4208
4276
4344
4412
4480
4543
4616
4685
4753
63
55
8
4821
4889
4957
'6025
6093
6161
6229
6297
6365
5433
03
6a
9
6501
6569
5637
6705
5773
6841
690S
6970
6044
0112
03
8 I ff 1
0.
4n27764IOI
76
4.774J 4813
76
552!)
5,5!)4
76
5:r.4
6338
74
7007
70rf2
74
?74:)
7823
74
^400
8,564
74
nw
9303
71
)m;Sj77ooi2
74
>70>l 0778
74
U4i)77i;>14
71
21 7r>
■i248
73
i!)OS
2981
73
3tM0
3713
73
137 1
4441
73
31(H)
6173
73
58;!)
6902
73
S6.)G
6029
73
ri8J
73,54
73
fiOO!l
8079
72
;^7;5!.) 7783i»;>
72
Wr2 0iVi4
72
)173!780;24o
72
Drtf)!)
0966
72
1012
1 684
72
i:{J9
i401
72
30I0
3117
72
3701
3832
71
4476
4646
71
5187
62,59
71
58'.)!)
785970
71
GOO!)
3680
71
7310
7390
71
8027
8098
71
8734
8804
71
0440
9510
71
0144
700216
70
0848
0918
70
1,5.50
1620
70
2iry2
2322
70
■imi
793022
70
3651
3721
70
4349
4418
70
6045
5116
70
5741
6811
70
6130
6605
69
7129
7198
69
7821
7S90
69
8.513
8632
69
9203
9272
09
98i"2'799061
60
0.580 i 800648
69
1266
1.336
69
i 9;)2
20^1
69
2637
2706
69
68
3321
3.!89
4003
4071
68
4635
4763
68
636.3
6433
C3
6044
CI 12
68
LOGARITHMS.
3"!
J'P N
6926
7603
o279
8963 1
9627!
810300
0971
1642
2312
806316
6994
7670
8346
9021
9694
810367
1039
1709
2379
a
806384
7061
7738
8414
0088
9762
310434
1106
1776
2446
812913
3681
4248
4!) 13
667^
6241
6904
7666
8226
8886
812980
3648
4314
4980
6644
6303
6970
7631
8292
8961
319644
820201
0868
1614
2168
2822
34/4
4120
4776
6426
819610
820267
Oi»24
1679
2233
2oS7
3639
4191
4841
5491
813047
3714
4381
6046
6711
6374
7036
7698
8368
9017
813114
3781
4147
6113
6777
6410
7102
7764
8424
9083
I81967C
8;0333
0989
1046
2299
2962
3606
4266
4906
6666
1319741
'820399
106;
1/10
2361
3018
3(i70
4321
4971
6621
826140
6787
7434
«060
8724
9368,
813181
3818
4614
6179
6843
6606
7169
7830
8490
9149
,806619
7197
7873
8549
9223
9896
810669
1240
1910
2679
806665
7333
8008
8684
0358
810031
0703
1374
2044
2713
806723
7400
807 «
8751
0425
310098
0770
1441
2111
2780
813247
3914
4681
6246
5910
667:i
7236
7896
8556
9216
319807
820,!64
1120
1776
2430
30«3
3736
4386
6036
5636
819873
1820630
11.86
1841
2496
3148
3800
4451
6101
5761
813314
3981
4617
6312
6976
0639
7301
7962
8622
9281
819939
820695
1251
1906
2660
3213
3866
4616
5166
5816
813381
4048
4714
6378
6042
6705
7367
8028
8688
9346
813148813514
82C004
0661
1317
1972
2626
3279
3930
4631
6231
5880
4114
4730
5446
6109
6771
7433
8094
8764
9412
326201
6862
7499
814 4
8789
9432
33t)0ilj830076
0u.)3 0717
1294 13,58
19341 19:).j
826334:826399
6981
7046
7628
7692
8273
8338
8918
B932
9661
9626
1130204
83026i
0346
0909
14S6
16.50
2126
2189
1826464
7111
7767
8102
9046
9690
826528
7176
7821
8467
9111
9764
820070
0727
1382
203.
2691
3344
3996
4640
6296
5945
4181
4847
5611
6175
6333
7499
8160
8820
9478
820136
0792
1418
2103
2756
3409
4061
4711
5361
6010
826693J326668
7240 7306
7886
8631 !
9175
9818
7961
669.
923i»
9882
330332jS303.'i6 83040o'830626
0973
1614
2263
'1 '"::;';? ''~^;?!'^^;°^' i'^^:il^^;'^a^764832828832892'832966 8
1102
1712
2.3S1
1160
1806
244
3276
0912
4640.
51 !s:)
5817
6461
7083
7716
8346
3338
39761
461 1 j
6247
6831
6614
71461
77781
8108!
3466
4103
4739
5373
6i);t7
6641
7273)
79011
8634
3630j
4166!
480 2 j
6437!
6071!
67((4i
7335'
7i)67j
86971
35931
4230 1
4.S66I
6,500 1
6134
6767
7399
8030,
8660
833083
64
3721
64
4367
64
4993
04
6627
63
6261
63
689,1
63
7626
63
8166
63
87;(6
63
339U38 839101 ;S39" 64:839227 83>'.S4
, 96671 97291 97921 9S66i <)M8
■340294 8,10367lo40420:8404.S) 84oij.ir,
1610
2236
2869
3482
4104
472d
1)09:
1:35;
23601
2983'
3606;
4229i
4h60
117:
1797
2422
3046
3669
4291
839362
9931
840603
1234
1360
2484
3108
3731
4363
49*'
•1
839116
840013
0671
1297
1 922
2647
3170
37hJ
4415
63
63
03
63
63
62
6'2
62
62
■ W4#
872
LOr.ARtTllMS.
i !
H
P I'
37
43
50
i'6
.'(lUjI
I
'2
3
4
o
fi
7
8
9
710
i84.^f)ns
6H:17
7573
R80")
!"419
OGlo
1'2
•24
3J
37
4y
4(i
56
t!
1*>
■2i
31)
3(i
4V
■k
54
84,)l(i0 84WJ.!
57 oO 6.S 42
851258
1870
2480
3090
3fi08
4301)
4913
5519
6124
0729
63;i:»i
70171
7034
8'25l
88'J6
9481
6461
7079
7096
8312
8928
9542
850095'850156
07071 0709
845284
5904
6V:3
7141
7758
8374
8939
9604
850217
0830
851320
1931
2541
3150
3759
4367
4974
5580
6135
678.?
351 381
1992
2602
32 1 1
3S20
4428
6031
6640
6345
6850
851442
2053
2003
3272
3381
4488
609/
5701
63ti(i
6910
6
845346
o9;)6
6585
7202
7819
8435
9051
96J5
850279
0891
845408
6028
6616
726 1
7881
8497
9112
9726
650340
0952
851503
2114
2724
3333
3941
4549
6156
6761
6363
6970
6
12
It
241
30j
3;i
41
47
63
6
1
1
23
29
3'.
41
4f.
6
12
1
23
2?
35
411
4S;
857332
7935
8537
9138
9739
860338
0937
1534
21
2728
857393
7995
8597
9198
9799
860398
0996
1594
2191
2787
857453357513
845470 815532
60901 6151
67081
7326 1
79131
85591
9174{
978S
850401,850462
10)4! • 1075
851564
2175
2785
3394
4002
4610
5216
662
6427
7031
6770
7388
8004
8620
9235
9849
851625851686
2236 2297
2816
3155
4063
4670
6277
68V2
6187
7091
2907
3516
4124
4731
6337
5943
6548
7152
8
846594
6213
0832
7449
8066
8682
9297
9911
350524
1136
845656
6275
6894
7511
8128
8743
9858
9972
850585
1197
351747
2358
2968
3577
4185
4792
6393
6003
6603
7212
80561
8657
9258
98^>!)
860458
1056
1654
2251
2847
8116
8718
9318
9913
860518
1110
1714
2310
2900
357574
8176
8778
9379
9078
860578
1176
1773
2370
2966
857631
3236
8833
9439
860038
0037
123G
1833
243(1
30
363323
3917
4511
5UU
6237
6878
7467
8056
864 1
863332'
3977
4570
6163
6755
6;Hii
693
7526
811
8703
;63l 12 863501
4036J
46301
5222
6814
6405
6996
7535
8174
8762
4096
4639
52S2
5874
6465
7055
7641
6233
8321
863561
4155
4748
6311
6933
6524
7114
7703
8292
8879
769 1
8297
8893
9499
860093
0697
1295
1893
24.S9
3035
857755
8357
8958
9559
800153
0757
13'
1952
2549
3144
I).
02
62
62
62
62
62
61
61
61
61
851809
2419
3029
3637
4245
4852
5459
6064
6663
7272
857815
8417
9013
9619
860213
0317
1415
2012
2603
3204
857375
8477
9073
9079
300273
0877
1475
2072
2663
3263
J
863620
4214
4803
54C0
6992
6533
7173
7762
8350
3938
3636301363739 863793^863368
4274
4867
6159
6051
6612
7232
7321
84U9
8997
4333'
4926
5519
6110
6701
7291
7330
8163
9056
4392
4935
557 R
6169
6760
7350
7939
8527
911
40 86923 :i36:l290]3693l9 S69408I
9813 987 71 9935 9994
87040 11870462 370521 370579
09891
15731
2156i
2V39
3321
3902
4482
750 875061
1
521
1017
1631
2215
2797
3379
3960
4540
1106
1690!
2273
2355
3137
4013
4593
1164
1748
2331
2913
34;t5
4076
4656
369466 869525
87UU53 370111
0633J 0696
12<;3
1806;
23391
2972]
0553
41341
4714i
12b I
1305
2443
30^0
3611
4192
477
869584 369642!
87017 0^870223!
075)
0313
133.)
1398
1 923
1031
2506
2564
3033
314!)
3669
3727
4250
4303
4330
4333
61
61
61
61
61
61
61
61
60
60
60
60
60
60
60
60
60
CO
60
60
4452
6045
5637
6228
6319
7409
7993
8586
9173
369701
370237
0372
1456
20 10
2622
3204
3735
4366
4945
369760
370345
0930
1515
2093
2631
3262
3844
4424
6003
375119 875177 87.V235 875293 875351
6640 5693 5756 " ' "
0213 6276 6333
6795 6vS)3 6910
■371 7429 7437'
7947! 3004 8062
8.5221 8579 8637
0096i 9153 921 1:
9669i 9726 9734 ^a^l\ Moitni wd.ju
!^. Ua' 8e02'l*S*0356 880 1 1 3I38O47 1 ,880528
375109
5937
656 1
7141
7717
8292
8:;66
9440
830013
875466
6045
6622
71:*n
7774
8319
8!)24
9497
880070
0642
875524
6102
6680
7 25 6
7832
8407
8931
9555
8S0127
0699
59
59
59
59
69
69
69
69
69
59
8755S2
6160
673
73i i
73S9
8464
9039
9612
88018
0756
59
59
58
68
53
58
58
58
68
58
68
68
68
5:3
6i
67
6?
57
57
57
I).
810056
62
6J7.J
62
6804
62
7511
62
81 28
62
8743
62
flgJS
61
997i
61
85008.)
61
1197
61
851809
61
2419
61
3637
4210
4802
5409
6064
6683
7272
807875
8477
9078
9079
SG027B
0877
1470
2072
2068
32(53
>!8G3368
14)2
3015
5037
6.!28
6S19
7409
7993
8086
9173
61
61
61
61
61
61
60
60
60
60
60
60
60
60
60
00
60
60
1
809760
59
7
870343
69
0930
58
6
1015
68
l'()98
53
■>
2081
58
I
3,'02
53
3844
58
6
4424
08
6003
58
4
87o5S2
68
>2
6160
68
>0
6737
68
< JU
C"
!2
7889
6i
)7
8464
67
n
9039
61
)0
9612
Ol
37
88018..
57
)9
1 0750
57
LOCiAlUTILMS.
373
1 — '>
PKj
N.j
6i0814
1
880871
2 j 3
4
5
6 1 7
8 1 9
!
UliD
880928 88093oi68l042
3.:31099i881100 881213
881271 881328
67
B
1
13^.
14 1,'
1499^ IO.jO
1013
!;i70
1727
1781
Mil
1898
67
.)
1900
2012
2009! 2126
2183
2240
2297
2354
2111
2468
67
3
2020
2081
. 20:wi
2090
2702
2109
2860
2923
2980
3037
67
w
1
•.i:,x'.
.3 100
32071
3264
3321
3377
34.34
3491
3048
3000
67
;'.■}
KJoi
3718
3770!
3832
o(^6"6
3940
4002
4009
4110
4i7i
67
14
()
42.M
4280
4342 1
43.)9
4450
4012
4009
4020
4082
4739
67
10
7
47:>.
4.i .2
4:i09j
4900
.0022 0078
5130
0192
5243
0300
67
i.j
Ci
<<8ji
5418
0474' 0031
0087 0!!41
0700
5757
.0813
5870
67:
.1
;;o
O'JJii
5983
60391 OOiiO
0152 0209
6205
6321
6378
61.'M
66
50
8dJ19i
880047
830o04 3;t;000
886710 8S0773
880829
880880
s;;o;ii2'88J'j98
1
7004
7111
7167
7223
7280
7336
7392
7419
7000 7061
6(i
1)
V
7 017
70; 4
7730
7780
7842
7898
7900
son
8007 8123
66
i;
3
sir;i
8v3o
8292
8348
8404
8400
«.,18
8o7:t
8'".29 8080
6(:
2'J
4
8741
8;','7
8803
8iio;)
8960
9021
9977
9i3!
9190
' 9246
6(;
2.H
t»
93;;')
9308
94141
9470
9020
9.',8.2
!t.i3M
9;i94
9700
98(/6
66
3i
9.'0!
9918
9974,3900.JO;39Uit.:'0:81"J141
8:30197
S0025:>
890309i8903ti.ol
60
31
7
j.ii) ; i 1
S90177 39O0r;3^ 0...i9 OSlOl O70O
0700
Oslv>
0868
09:4
60
4.I
S
OJ.i.l
1030 1091 i 1147 1203 12.59
1314
1370
1420
1 182
60
.)()
1.037
lr,!';i 164)
1700J 176i» 1810
1872
1928
I98;i
2039
6(i
5">
>9Jo;io
8i)21>j:;l!.'220o'
892202 892317 892373 892429,8.0244.1
892040'3.)2090
(;
1
■ liOl
2;i)/
27621
2818 2873
•J929 .VJi,
3040
30!!0 3101
60
1 1
•2
uiOj
'.j26i
33 IJ'
33;3 312.)
31>i 3010
3090
3001 3700
60
1;
d
a .■>;:
317
3.J73
3928 3 181
4i»3:) 4091
4100
4200 4201
60
.i_.
4
131. i
43; 1
4427
4182
4..38
4..93 4018
4701
4709 4814
6f"» '
•)•
•; iT.i
4J20
4980
0030
5091
0140 0201
0207
0312;
5307
6i\'
',i'i
U
1:3
0478
6533
5088
0644
0099 0."04
5809
5.J01
5920
55
•f
(
:,'■;.,
OO.iJ
OCH )
0110
6190
02011 0300
G301
0416
6471
53
11
(?
orJi!
0,..ll
Oii.iO
6092
0717
0802 0J07
6912
6;)07
7022
50
Vi
y
7i).'7
7132
7187
7212
729i
7302 7 107
7iJ;i
7017
7072
65
i'.ii)
i:)10i:i8y;iiv*.!k;:*r;3r ><37792:Lii)78l7;897;!02l8y7^J0?Wf;8!)12
89_;007898t22
50
■ 1
81i!v tjJ3l
Hii'i 8:>. 1 !
839.4; soil H.,0!)
8'.ri!
.■^010
8070
(").'
l!
'2
«;',j 87^0
S:\ii
889(1
894 1 j 6;»99' 9._)04
9'. 09
9104
9218
6c.
i;
'■i
;)^7;i; 9328
95J3
9137
9492i 9)47] f!;!;)2
9000
9711
9706
66
2'
^
'.i:<}\\ 9Ji"0
9930; 9980
900{)39i90i);)91^S0ii 1 49I900203
9002'S 900312
56
2;
tt
J<)o:!Jti9:;..!122
9004;0,:);r).j31
008J
OOtOI 0090) 0719
08041 0809
1.
3;.
a
u:m3 0.)08
HHi lo7T 1131
11861 12401 1290
1319
1104
t
lto..'l ni;.
l.O; iti2ii UiTo
ivai i780j 1.^10
l.'504
1918
54
41
V.
•.i!iii3 2i).il'
2M2i 2100: 2221
2270 vi329 2384
2438
21!.'
64
4;,
,0v
2)i;j ji.ii
20j;i 27I0 2764
281.81 2;'i73 2927
_' _ 1
29 U
3036
54
.(.;i..'.'j{i!yo3i44
;9U3i99
9032.^:!'0330;'!.'0;J301 190341 0i903470
9j3024'90357y
.»
1
;;.i;;.i; 3ij87
;;741
3;90i 3819 3;'94j 3.108 4.)12
1 4000; 4120
64 i
li
■^
417i! 42 .>a
4283
43371 X:)^i 4 140 4t99 4003! 4607
4001
54 1
i>;
;i
47u;! 4." 70
4.'24
4J,8i 4.13 i 4.183! .0040 00941 OILS
5202
54 !
•) J
4
O.'OO.j .03 10
o;i;i4
04I8 0472 5i26i ovJO .^.'■:34l .0<3.i3
.57 12
54 i
•>:
;i/:*.;i d>i'i\i
5904
o9.)8i 6'JI2 0)v)S! o;i'<»i OTS 0227
6281
64 ■■
a.
h
i>.Ji.); 03:^91 «t43
o;9;j 0001 j o^ioii (.'Oo^i 6:12 0.33
0;i2n
54
3 .
7
n.i\ t;!;'2i 6,).>i
7030! 70.19; 7li;j 7"! Oi 720 7;v:)4
730fa
54
j;i
ij
74it 7ij.Ji 7.(1;)
7.>;3 7'i.'tt ;,»8;>l i73I 77.d7: 78il
7.89S
64,
\ 4.
1
i
T:i;.4; t,.;ii)2i 8;j.,<i
8110 s>iti,ii sii7 :;2r0 S3:i! 837s
84;il
54:
Ut
^(08 l;!0 90.l.j:i9 ilvJd.yj
lP(\li w90..Ji99'>' ,7.>.'. 908:.>7'9!i88y0,;K).J94
uoo.is;
•M !
1
9l):.'li 9J71i 912..
918! 1 9i.i.;' . ■...■'' 9312J OS"*!! 9419
9.00?
1 841
11
V
<i.V)U. ;..!,)! 9M],i
9,l»: 9770 P.JJ 9877 W.<m *t9:;i
rtiool'w
65
l!^
. '!(■!;.>! 9!(_'J44'!H'J!?r
91i'.:rl Ol(..'5i!4 9l030'':9ir:l! WUa4. 4 li!0 18
0,0?]
53 .
VM
4
0324j miS\ 0(31
! 07>^4 083~, O^i'lli Q914! 0998i lOOi
iio.
\ 51 1
*»T
lJ.'*«i 12)li 120i
13.17 '..on
(.jii, ii:? 1030 1084
UiA(
P^
'.i\>
t
1090: i;4:'
1 1797
la'Ji VMy
19)61 2;).)9 200:} 21 Ih
SI'OO 63
37
7
222i 227.5
■^^■■», ^aa^
I 24.}
( 2488 2011; 2ii9^
1 264i
2700 a J
42
I
270.S 280t
28.Vjl (JVM.
i 290(
) 3019 39 72' 312.
) i}T\
* 323t M
4i
!>
;»2M.i ;«3.>7
;n9^
) Ml
i 349i
i :04i'. 30'>* 3».'..
. .37'vW>l ;l7til '«
^^
■
n H \
II ih
m<
ii
874
LOGAIUTUM9.
l^p
N.
{ t
*2
a 1 4 1
5 1 6 1 7 j 8 9 1
U.
330
)l3814;9139(J7i913920i913073 91 1026i91407i) 914132914181 914.;37j!)l429o
53
6
1
4313
4396
4449i
4502
4555
4608
4660
4713
4766
4819
63
11
2
4872
4925
4977
5030
5083
5136
6189
5241
6294
5317
53
63 *
16
3
5100
64 .3 ■
6505
6568
5611
5664
5716
6769
5822
6875
31
4
5927
5980
6033
C085
6138
6191
6243
6296
6319
6401
53
27
o
0154
6507
6559
6612
6664
6717
6770
6322
6875
6927
53
32
0080
7033
7085
7138
7190
7243
7295
7343
7400
7463
53
37
7
7506
7558
7611
7663
7716
7768
7820
7873
7925
7078
63
4i
6
8030
8083
8135
8188
8240
8293
a:i45
8397
8450
8502
62
48
9
830
8555
8607
8659
8712
87 64
e816
8869
8921
8973
9026
52
919078 919130
319183 919235 919287
119340 919392J
9194441919196 919649
52
5
1
9(J01
9653!
9706 9758 9810
9802
9914
9967 920019 920071
62
10
2
«0I23
D20176
W0228 920280 920332
J20384I920I36
920489
0511
0593
62
16
3
0615
0()97
0749
0801
0853
0906 0958
1010
1062
1114
62
21
4
1166
'218
1270
1322
1374
1426
1478
1530
1,,82
1634
52
26
I,
1686
1738
1790
1812
1894
1940
1993
2050
2102
2151
62
31
ti
2206
2258
2310
2302
2414
2466
2518
2570
2622
2674
52
30
7
2725
2777
2829
2881
2933
2985
3037
3089
3140
3192
62
42
8
3241
3296
3348
3399
3451
3503
3555
3607
3658
3710
62
47
840
3762
3814
3805
3917
3969
4021
4072
4124
4176
4228
52
924279
924331
924383
924434
924486
924538
924589
924641
J24693
924744
02
6
1
4796
4848
4899
4951
5003
5054
5106
5167
5209
6261
52
10
2
5312
5364
5415
6167
6518
5570
5621
5673
6725
5776
62
10
3
5828
5879
5931
5982
6034
6085
6137
6188
6240
6291
51
20
4
6:542
6394
6115
6197
6548
6600
6051
6702
6754
680..
51
•ao
5
6S.)7
6i)08
6959
7011
7062
7114
7165
7216
7268
7319
61
31
6
7370
7422
7473
7524
7576
7627
7678
7730
7781
7832
51
36
i
7883
793.)
7986
8037
8088
8140
8191
8242 8293
8345
51
41
8
8396
8417
8198
8549
8601
8652
8703
8754 8.H>5
8S57
51
46
850
8908
8959
9010
9061
9112
9163 9215
9.>«6 9317
9368
61
929419
929470
929521
929.j72i929623
929674
929725;929776929827l929879
51
6
1
9930
9981
930032 930083:930134
930185
9302361,930287
930338 930389
51
10
2
930440
930191
05421 0592
0643
0694
0745
0796
0847
089.i
51
15
3
0949
1000
1051
1102
1153
1204
1254
1305
1 356
1UI7
51
20
4
J 458
1509
1 560
1610
1661
1712
1763
1814
1865
1915
51
ae
5
1966
2017
2068
2118
2169
2220
2271
2322
2372
2423
51
81
6
2474
2524
2575
262b
2677
2727
2778
2829
2379
2930
51
36
7
2931
3031
3032
3133
3183
3234
3285
3335
3380
343,
51
41
8
3487
3538
3589
3639
3690
3740
3791
3^41
3m92
3913
51
46
9
3993
4044
4094
4145
4195
4246
4290
4317
4397
4441.
51
50
860
93449b
934549
934599
934650
934700
934751 i934S01
934852
934902
934953
5
1
5003
5054
5104
5154
5205
5255
530fc
5356
5406
6467
50
10
2
5507
5558
6603
5G5fc
5709
5759
6S09i 5860
5910
696!
50
15
3
6011
6061
6111
61 6i.
6212
6262
63ir
! 6303
01131 046;l
60
20
4
6514
6564
6614
66ac
1 6715
6765
681,
6;iti)
6.416
696ti
50
35
5
7010
70';'o
7117
7167 7217
7267
7317
7367
7418
746.
50
30
6
7oifa
756o
7618
7068 771ti
7769
7811
) 7d69
7919
796!
50
3S
7
801!
806!
8119
81G9i 8219
826:.
832(
) 8370
81^0
8471
50
4€
h
852iJ
8570
8620
86701 8720
877 (.
b820 88 ;(
892U
897t
50
4:
fl
87(i
9021
9O70
9120 9170 9220
927(
9320 9369
9419
9;j£t91fc
946:
50
,9395U
!93956S
9396191939609 9397 1993976!
939819l93986?
93996!
i 50
t
1
940018 94O0b.^
940118;.>4o;68!940218;940267
9403i79'U)367
940417
940467
50
If
1
0516 056(
06 ic
0666; 07 It
) 07 6r
OSlol 086:j
091;
096
50
li
S
1014 1061
nil
11631 1211
i 126a
13 is! J26,
Ml.
146
' 50
21
"1
1511 1561
1611
IWiO 171<
)l iTdC
180
1 I85b
190 J
195.
i M
2.'
i
' y)08 205E
! 2107
2157 220"
r 225t
230
3 235r
240i
• 245
. 50
sr
(
) 2504 255')
1 260r
2653 270;
J 275i
280
1 2851
2001
2951
J 60
3f
'
300(
) 304f
> 3091
» 3148 319i
< 3247
329
7 334t
339t
) 344.
) 40
4{
f
i 349t
) 354'J
1 3591
( 3643 369.
I 374i
. 379
i 3341
3S9l
) 393<
) 49
4i
t
1 398J
) 4U3i
1 4085
) 4137 4jSt
J 423c
■«N— (••fir—
a 433i
. 438'
I 143.
i 49
9
U.
■i9l4M0
63
J
4019
63
I
5317
63
i
6876
63 •
)
8401
53
5
6927
63
D
7463
53
")
7078
62
I)
8602
62
3
902'J
62
B!
119649
52
')<
W0071
62
1
0r,<)3
62
2
1114
62
}
ie34
52
2
21 I
62
•i
2674
52
3192
62
ci
3710
62
6
4228
52
3
924744
62
9
0261
52
5
6776
62
6291
61
4
680,.
51
H
7319
61
1
7832
61
3
834o
61
>i>
8S67
51
7
!):i68
61
!7
929879
61
)3l9;{n389
61
t7
OSO.i
r>\
)(i
IUI7
61
3o
1916
61
?2
2423
51
r9
2930
61
HO
343,
.^1
■J2
39 13
.^l
J7
4441.
61
60
0:2
934963
OH
6467
50
10
696!
511
i3 cum
60
It
696ti
50
lb
746.
60
!i
796!
60
■^fl
a47(
50
■M
897t
50
IS
946:
60
It
93996t
i 50
17
94046;
60
I;
090
60
1
116
' 50
o;
196.
i 60
U
> 246
. 50
296(
J 60
9t
) 344.
) 40
.9(
) 393*
) 49
13'
1 143.
i 49
LOOAPJTHMS.
1 *"
881
t> , 1 i 2 1 3 1 4 • 1 r. 1 6
7 1 8 1 9
49
91418,'. 941632 94l.V^I '944631 944680 9447291944779
944823:944877 941927
r
1
4976
6026
6071
6124
517a
6222
6272
5321
5370
5419
49
10
2
6469
651b
6667
6616
6666
6716
6761
6813
5862
5912
49
IS
3
6961
6010
6059
6108
6167
6207
6266
6306
6354
6403
49
20
4
64)2
6..0I
6661
6600
6649
6698
6747
6796
6845
6894
49
36
6
6i'43! 6992
7011
7090
71^0
7189
72.38
7237
7338
7386
49
29
6
74341 7483
7.i32
7681
7630
V679
7728
7777
7826
7878
49
34
7
79i,l
7973
8022
8070
8119
8168
8217
82C6
8315
8361
49
39
H
8U3
8462
8611
S660
8609
8667
8706
8766
8801
8863
49
44
9
8902
8961
8999
9048
9097
9146
9196
9244
929*2
9341
49
49
890
94939i)'9494.39
i949,,88
949630i949586
949634
949683
949731
949780
949829
5
1
9878 •9926! 9976
960024
960073:960121 1960170
960219
950267
960316
49
10
o
96036)i950114!9j0'l62
0611
0660
0608
0667
0706
0764
0803
49
18
3
0861
0900
0949
('.'"97
1046
1096
1143
1192
1240
1289
49
20
4
l;i.38
1386
1436
J 483
1632
1680
1629
1677
1726
1776
49
34
6
1823
1872
1920
1969
2017
2066
2114
2163
2211
2260
48
29
6
■2MS
2366
2105
2463
2602
2660
2.699
2617
2690
2744
48
34
7
2792
'i841
2839
2938
2986
3034
3083
3131
3180
3228
48
39
8
3.'7(»
3326
3373
3421
3470
3618
 3666
3616
3603
.3711
48
44
3760
3803
3856
3905
3963
4001
4049
4093
4146
4194
48
48
900
961243i9,'>4291
954339
954387
9.54436
964434
964632
964680
9546281964677
5
1
4;2.,i 4773
4821
4869
4918
4960
6014
6062
5110
5158
48
10
2
6J07
5266
6303
6361
0399
5417
6496
5643
5692
5640
48
14
o
COS.S
6736
5784
6832
6880
6,928
6976
6024
6072
6120
48
19
4
616S
6216
62G")
6313
63iil
6 109
6467
6606
6563
6601
48
24
6
0iJ49
6697
6746
6793
6840
6888
6936
6981
7032
7080
48
29
6
7128
7176
722 1
7272
7320
7368
7416 7464
7612
7659
48
34
^
1
iiior
7656
7r03
7761
77!)9
7847
7894 7942
7990
8038
43
38
S
8086
8134
8181
8229
827 7
8326
83? 3 8421
8468
8616
48
43
9
s)IO
b)6l
969U41
8612
S6.^9j 87071 8706
8803
88,i0 8898
8946
8994
48
48
969089
9,>91.37 969186
969232
959280 959328'969376
969423
9.59471
5
1
9,. 18
9>fl6
9614
9661
9709
97.67
9804
9862
9900
99.17
48
9
2
i'ni'.)!} 960042
969090
9001.33
960186
960233
960281
960328
960376
960423
48
14
3
960171 0,il8
0666
0613
0661
0709
0766
0804
0861
0899
48
19
4
0916
099 I
1011
1089
1136
1184
1231
1279
1326
1.374
47
24
6
1121
1469
1616
1663
iOll
16..8
1706
1763
1801
1348
47
28
6
1896
1943
1990
2038
2'i86
2132
2180
2227
2276
2.322
4'r
33
7
2369
2417
2464
2611
2669
2606 2663
2701
2748
2795
47
38
8
2813
2890
2937
293;)
3032
3079
3126
3174
3221
3268
47
42
9
9;;()
3310
33G3
3110
34,67
3.601
3652
3599
3616
3693
3741
47
963788
963836
903882
963929
963977 964021
961071 961118
964166
961212
47
£
1
■1 JOO 4307
4304
4401
4148
4496
4612 4690
4637
4681
47
S
2
4731 4778
4826
4872
4919
,1966
6013 6001
6103
6 1 56
47
i 14
3
6202
6249
6296
6313
6390 6137
64,84 6631
6.678
6626
47
19
4
.0672
6719
6766
6813
5860
5907
6964 6001
6048
6096
47
23
6
6142
0189
6236
6283
6329
6376
6423 6170
6617
66('4
47
29
6
O.ill
t;6,i8
6706
0762
6799
6346
6892 6939
6986
7033
47
. 33
7
7080
7127
V173
72,;0
7267
7314
7361 7408
71,5!
7.501
47
38
8
7,348 7,i96
7642
7 688
7736
7782, 78291 7876
7922
7969
47
42
9
80 1 6 S,)62
8109
8166
8203 824:){ 8296; 8313
8390
8436
47
9684831968630
96,'^,;76 968623
96867096i7i6'968763 96881
968866
968903
47
6
1
MV,0 8996
9043
9090
OUlol 9183
9229 9276
9323
9369
47
9
2
9116 9163
9609
9660
9602 9649
9696 9742
9789
9836
47
14
3
9382 9928
9976
970021
970068 97011 !
970161970207
970264
970300
47
io
4
970347
970393
970440
0486
Ou.iH
(',y/9
06'j6; 0672
0719 0766
46
29
5
0812
0368
0904
0951
0997
1044
1090 1137
1183
1229
40
23
6
1276
1.322
1369
1416
1161
1.608
1. 6.541 1601
1647
1693
46
32
7
1740
1786
1332
1879
1926
1971
2018 2064
2110
2167
46
37
8
2203
2249
2296
2342
2388
2134
2481 2527
2.573
2619
46
41
9
2666
2712
2768
2804
2861
2397
2943 2989
3035
30t.2
46
!J i
i! iil
iii^
. .
i 1
,t I
J76
LOOARITHMS.
5
14
IH
37
41
6
9
14
18
2:»
•i7
32
3t<
41
N.
340
1
2
3
4
900
J
3
4
6
6
7
9731iS
30D0
4051
451i
4972
f)432
5891
63f)0
fisoa
72aa
1)77724
8181
8037
9093
9548
980003
0458
0912
13fifi
1819
1
973174
3()36
4097 j
45581
60181
6478
6937
6396
6854
7312
973220
3f.82
4143
4604
6064
6524
6983
6442
6900
7358
973266
3728
4189
4660
6110
5570
6029
6488
6946
7403
977769
8926
86u3
9138
9594
980049
0503
0957
1411
1864
977815
8272
8728
9184
9639
980094
0549
1003
1456
1909
977861
8317
8774
9230
9685
980140
0694
1048
1501
1954
973313
3771
4235
4696
5156
5616
6076
6533
6992
7449
977906
8363
8819
9275
9730
980185
0610
1093
164'
2000
977952
0409
8865
9321
9776
980231
0685
1139
1592
2045
977998
8454
8911
9366
9821
980276
0730
1184
1637
2090
982452
2904
3356
3307
4257
4707
6157
6606
6055
6503
982497
2949
3401
3852
4302
4752
6202
6651
6100
6548
8
9
973369,973405 97345 1 973497
3820 3866 3913 3959
4281 4327 4374 4420
4742 47 88 4834 4880
5202 6218 6294 6340
6062 6707 6763 6799
6121 6167 6212 6258
6579 6626 6671 6717
7037 7083 7129 7175
7495 7541 7686 7632
978043
8500
8956
9412
9867
980322
0776
1229
1683
2136
973543
4005
4166
4926
6386
5845
6304
6763
7'?20
7678
978089
8546
9002
9457
9912
980307
0821
1275
1728
2181
9SG951
7398
7845
8291
8737
91 S3
9628
990072
j 0516
0960
982543
2994
3446
3897
4347
4797
6247
5696
6144
6593
982583
3040
3491
3942
4392
4812
6292
5741
6189
6637
982633
3085
3536
3987
4137
4887
8337
6786
6234
668i
978136
8591
9047
9503
9968
980412
0867
1320
1773
3226
982678
3130
3581
4032
4482
4932
6382
6830
6279
6727
936996 987040;
7443
7890 j
8330
8782
9227
9072
990117
0561
1004
748S
7934
b;>jl
(■^S20
9272
9717
990UU
0605
1049
9870S5
753«
7979
8425
837 1
9316
9761
990206
0650
1093
1802
1846
1890
2244
2283
2333
2686
2730
2774
3127
3172
3216
3568
3613
3657
4009
4053
4097
4449
4493
4537
43B9
4933
4977
5328
6372
5416
937130
7577
8024
8470
8916
9361
9306
990250
0694
1137
45
45
46
45
45
45
45
45
45
45
937176
7622
8068
8514
8960
9406
9360
990294
0738
118!
5 991359 iiOl 103,991448991492
1935
2377
2819
3200
3701
4141
4531
5021
5460
67 995811 995854 995898
6337
6774
7212
6205
6249
6293
6643
6637
6731
7030
7124
7168
7517
7501
7005
7954
7998
8041
8390
8434
8477
8326
B869
8913
9261
9305
9348
9096
9739
9783
8086
8521
8956
9392
0826
991536
1979
2421
2363
3304
3745
4185
4625
5065
5504
995942
6380
6818
7255
7C:)2
8129
8564
9000
9435
9870
991530
2023
2165
200
3343
3739
4229
4669
6103
654'
1
995986
6424
6662
7299
7"36
8172
8608
9043
9479
99i8
45
45
45
45
45
45
44
44
44
44
991626
2067
2609
2901
3392
3833
4273
4713
6152
6591
996030
6468
6906
7343
7779
8216
8652
903
9522
9967
44
44
44
44
•11
44
44
44
44
43
A TABLE OF S(iUAHKS, tj0hK9, AND ROOTS.
377
1
9
D.
n
73.543
46
4005
46
41t)b
46
4926
46
6386
46
5846
46
6304
46
6763
46
7'?20
46
7678
46
i
)78136
46
8591
46
>
9047
46
9503
46
I
9908
46
r <
)80412
46
0867
46
>
1320
45
i
1773
46
1
3226
46
J
982678
45
5
3130
46
i
3581
46
?
4032
45
7
4482
46
7
4932
46
7
6382
45
6
6930
46
4
6.i79
45
2
6727
46
987176
45
7
7622
45
4
8068
45
(»
8514
48
6
8960
45
1
9406
45
6
9360
44
990294
44
4
0738
44
7
1182
44
to
991626
44
'3
2067
44
>5
260S
44
)7
2901
44
H
3393
44
3.0
3833
44
29
427f
44
jt)
47K
44
)H
61oV
44
t7
5591
44
it
99603(
44
i<l
646t
44
j'J
690(
44
3£
?34;
44
It:
; 1 r
•ii
?v
. 821(
44
Bt
i 866
44
4J
1 908"
44
7i
t 952
i 44
li
t 996'
r 43
No.
flqonre.
Cut. p.
Sq. Root. Cuho Iloot
No.
64
Sqiiaru
C'tilie.
.^q. Root.
Cube Root
I
1
1
10000000 1 000000
4096
262144
80000001)
4000000
a
4
8
14142136 1260021
66
4226
274626
80622677
1020726
3
9
27
l7320.')08 1412260
66
4366
287496
81240331
4041240
4
16
64
20000000 1687401
"B7
4489
300763
81863628
4061618
6
25
125
22360680 1 709976
68
4624
314432
82462113
4081656
6
36
216
24494897 1817121
69
4«61
328609
82066239
4101660
7
49
313
26467613 1912931
70
4^)00
343000
83666003
1121286
8
64
512
28284271 i 000000
71
6041
367911
84261498
4140318
9
81
729
30000000 2080084
72
6184
373248
848.52814
1160168
..
10
100
1000
31022777 2164136
73
6329
389017
86440037
4179339
11
121
1331
33106248 2223980
74
547<)
40.5224
86023263
4193336
12
144
172S
34011016 2289128
76
5626
421876
86602640
1217163
13
169
2197
3 6066513 2 361. 3;<6
76
6776
438976
87177979
4i368J4
14
196
2744
37410574 2410142
77
6929
466633
87749644
4254321
16
226
3376
38729833 2 466212
73
6084
474562
88317609
■1272669
16
266
4096
40000000 2519842
79
6241
493039
80881944
4290841
17
289
4913
41231066 2571282
80
6100
612000
89442719
4308870
IS
S54
6832
42420407'262'>741
81
6661
631411
90000000
4 .326749
19
861
6859
43688939 2608402
82
6724
661368
90663861
4344481
20
400
8000
44721300 2714418
83
6889
671787
911043.36
4362071
21
441
9201
46826767,2763924
84
8,^
7066
692704
91661614
4379619
22
484
10048
4 09041 58'2 803039
7226
014126
9219.6416
4 .396830
23
629
12167
4796831612843867
86
7396
636066
92736186
4414006
24
676
13824
4 8989796 ,2 334499
8T
7669
668603
9327.3791
4431047
26
026
15626
&0000000!2924018
83
7744
681472
9380.3316
4447960
26
676
17676
60990196;29d2496
y:)
7921
704969
94339811
4464746
27
729
19633
51961524 3000000
90
8100
729000
9436.3330
4481406
20
784
21962
6 2916026'3 036689
91
8281
7.63671
96.393920
4497941
29
841
24389
63361648 3072.'il7
92
8461
778688
9 .69 16630
4514367
30
900
27000
64772266 3107'232
93
8619
804367
96436608
4530656
31
961
29791
66677644 3 141.381
94
S336
830584
96953697
46468.36
32
102{
32768
66668612 3174802
96
9026
867376
97467943
4662903
■
33
1089
36937
5 74 16626 3 207634
96
9216
884730
97979.590
4578867
!
34
1166
39304
68309619 323961J
97
9409
912073
98488678
1594701
85
1226
4287;.
69160798 3271006
93
9604
941192
98991949
4610436
.S6
1296
46666
60000000:3301927
99
9801
970299
9949374 1
4  626066
.Hi
1361)
bmr,.\
6 0327026;3 332222
100
1 0000
1000000
100000000 4641689
3S
1444
64872
01614140,330197;>
101
10201
1030301
100498766 4667010
:!0
1621
69319
6 244 9980:339 1211
102
10401
1061208
100996019!4672329
40
KiOO
64000
6 32 15663 3419962
103
10609
1092727
101488910 4687643
■!1
Hi 3 1
68921
64031242:3418217
104
10:ilt)
1124861
10 19.303904 702669
42
1764
7408d
64807 107'347C02V
106
11026
1167li2.5
10 2469,)03!4 717694
43
1849
79607
6667438613603398
106
11236
1191010
10 29.56301 ,4 732624
44
1936
861. HI
6 633249ij!3 630348
107
11149
1226043
103440804:4747469
\>
2026
91125
6708203;)l3666y93
108
11664
1269712
10392304«,4702203
• UJ
21 Id
9; .WO
07823300i36830Ks
109
llsSl
1295029
10 44030o6'4 7768.06
■'
4<
22i.i'J
10;5823
68666546!360c;820
110
12100
1331000
I048808o6i4791420
•1.!
2304
\W„'H
6 .'282032 '3 634241
111
12321
1367631
106366.533:4805396
49
2101
117619
70000000 3669300
112
12641
1404928
10 .60300.02:4 820284
60
2600
12u000
7 07 10678:3 6840.) 1
113
12769
1442897
i0630!463:4834.6S8
61
200 1
132661
7U11284!37081:;o
il4
12996
1481614
106770783 4848808

62
2704
I40o0w
72111026:3732611
116
13226
1,520876
107233063 4862944
b3
2;:0a
Mo.ii?
72301099;376628i)
116
13466
1660896
10  77, ;3296 4870999
!
,j4
2;il6
16,4o4
73484690 3779763
117
1063'.)
1601613
10816:) 538 4890973
66
3U2;i
16637)
741619Su!3802963
118
13924
hi 43032
108627306 4904368
A
.i^>
3i:>6
176616
748331 is;3 826802
119
ilOl
168^169
10908712!:4918686
.)7
3249
l.>361.93
76498341;3843.601
1.H)
1 ! 100
1728000
109644612 4932424
k
;i:H
33(jt
!9.M)J
7fi!6773!!3.870877
i2'
1 4641
177166!
11 0000000 4 94:>0a3
>
69
34t;l
2063;9
7 681 1457 Is Ly 2900
122
1488 4
1816848
1101.53610 4969676
1
lit;
3G00
216000
774696671391 1867
123
16129
1 860867
110110636)4973190
1
(5!
3V2i
226'J81
7 6102497
3 93649 r
3967802
124
16;i76
I;i0u624
111366287 4986631
C2
3844
23.':!328
78740079
126
1602o
1963126
111303399,6000000
?
63
3989
260017
79372539
39790.57
126
1637f
2000376
ll224972'i,6 01 3298
1
i'i
!
l!f
h
J r;
nil
'I j I '
M i
'I i
! (
373
SQUARES, CI UKS, AND U00T8.
No.
1Q7
Sqiinrt.i C'lilic.
130
lai
133
131
135
i:<6
137
138
139
MO
141
142
143
144
146
146
147
148
14!)
160
151
162
163
154
155
ISO
167
158
169
160
Itil
162
163
104
10.'
166
137
168
169
170
171
172
173
174
175
176
177
178
179
ttiO
181
182
183
1 84
18o!
l8o
187.
18H
189
1U120
I63H4
Itilitl
16900
17101
17424
176891
179661
18226
184961
187091
19014
193 Jl
196901
198811
20164!
2(J44i»
20736
2102";
21310
21009
21004
22201
22.>00
22?01
23104
2340!)
23716
240iJ.'
24336
24649
24964
36281
20600
26921
26244
26;j()9
26896
27226
276)6!
2788;)
28224
28061
28900
29241
29684
29929
302Vii
30026
3ii!i.'6
313J9
3168i
3J011
3'4iJ0
32 ,"1)1
331
2048383
2097162
2146089
2197000
2248001
2290968
2362ti3
2400 1 U4
2lli03/.,
2616 i:o
2671363
2028072
2686619
2MU)00Ul
2riO:U21'll
286.1'28,si 1 1
3:iis9
33860
34226
34.".96
3 1909
36341
36721
I
8q. lUiut. Ciitij lluul
11 2694277 1602H620
1 131 37086! 5 0396S1
ir367H167i.»0627;4
ll40i;6436066797
11 4456231 1 6 0787 63
ir4!i;912636091613
llo3266.'66
116768369 6
104169
117230
■ 1^9928
1 12603
•16613;
29:^ 1207
298,)98 4
30 (8626
3112136
3176623
32ll79i
3307949
337..000
344296!
3611808
3681677
3662264
3723876
3796116
386i)»93
39443 1i
4019679
40ll(i000
4173281
4261628
4330747
4410944
4492126
6 189600 1 6
601903h;6
7046999:5
7473414 6167649
78982616 18010
8321696;.) 192494
8713421 6 204828
9l637.")3!6217103
1 1 968260 7 1 6
120000000 6
I 204169 lo! 6
1208.'J0460'6
^39321
211 183
263688
206
N.»
100
191
192
193
194
196
196
197
198
199
200
201
202
203
■201
20)
06
207
208
flquurc
637209
i21243.6676'2776:i2l2' "
16.V)i6l6289.<72
2066666 6 .3014.)9
122474487 6313293
1 2 288206t;6 326074
1232882S0 16336803
123693169;6318181
l2409673i)!636010d
12 4498996; 6 37 1 686
124899960 6383213
126299641 63i)4691
1 2 .6638061 6406120
126096202, u417601
126401106 6428836
l.'6886776644012
127279221 i6461362
1276714.636462666
12 8062486 ;6
Cnbf.
1284.")232rti5
46742..»6;128840987a
4667433! 12 9228430 15
4M1032;12961481 t 6
4826809, 13 0900000 '6
4913000!i30384048"6
47.3701
48481)0
496866
606879
617818
5287 7. '5
639668
6000211! 13 076696»,6 6^0 199
■56129,;
(7 206^1
■ 604079
5088448 131148770
51777171 131629464
526 J024 1 3  1 90.9060 6  ,)82770
5369376 1 :', • 228766615 f 934 46
6461776: 13266 4992:6
5.6462:j3J33041:)47i6
66:i;l7.)2; i 3 ■ 34 1 664 1 6  62.)226
6736339 1 3  3790882 5  6367 I i
[>:;32O00 13416 4(179 6646216
6929711 1346:i62.10;66)6661
(;;)28608' 1 3  4907376 6  667061
6 l2'U87j 13 6277493:6 6771 11
6229.JO 4' 1 3  6646600:6  687734
63;'.1626l.'.601 '17066 OOoOlO
6434866 13638181715708267
6.j39203 130747943;67 18479
66446 72: 13711 3092;6 728664
0751269 137477271o738794
10
!11
212
213
214
215
216
217
213
219
220
221
222
223
224
225
226
227
228
■229
230
■231 i
232l
233
234
236
236
237
2W
239
614673 210;
24 1 ;
242!
243 j
2441
246 1
246
247!
•248 1
249
260
251 1
252!
36100
36IB1
36864
37249
37636
38026
38416
38809
39204
39601
40000
40401
4t)804
41209
41616
42026
42436
42349
43264
43681
44100
44621
44944
45369
4.6796
46226
46666
47089
47624
47961
48100
48841
49284
497^9
60176
50626
61076
51529
51984
62441
.52900
63361
53324
61289
54766
56226
06696
56169
66641
57121
57600
58081
68664
69049
59636
60026
60616
01009
61.604
62001
tf2600
63001
63504
Sq. Iloot. Cub* Root
68590001137840488,5
09678711 13 •8202750 5
70778881138561066 6
71890,17 138921440.6
7301384 13n283883 5
7414876 139012400 5
7620636 U'OOOOOOOl.5
76 4.6373 1403.jfl888;5
7762392 I407I2173'5
7880)99
8000000
8120601
8242 108
8366427
848966 4
8616126
8741816
8889743
8998912
9123329
9261000
9393931
9.62812S
9663697
9890341
993S375
10077696
10218313
10360232
10603469
10648000
10793861
10941048
11089667
11239424
11390626
11.543176
11097083
11862362
12008939
12167000
12326391
12487168
12619337
1281290 4
12977875
13I442.J0
13312063
13431 27 J
13661919
13824000
13997621
14172188
14348907
14526789
14706126
14886936
16069223
16262992
16438249
15626000
16813251
10003008
11
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
U
14
14
14
14
14
14
14
IS
IS
16
15
16
15
16
16
16
I5'
15'
15'
16'
IS
IS
16'
15
15
15
15
15
15
15
lo
16
15
15
15
1 067360 15
14213666
1774469:5
2126704 5
2478068:5
2828669 i5
317821 1 15
352700115
387494615
4222051 5
4,56832315
4913767
6258390
6602198
.59451it5
6237388
6628783
69693t
730<)199
7648231
7980436
8323970
8660637
899(i644
9331815
■9666296
0000009
0332961
06fi6192J6
• 0996689 i.i
1327460
1657609
1986842
2315402
2643375 J6 •
•297068616
.3297097 6 •
3622916 6 
3:11801316
4272486 16 
4696248 '6 
491933 I '6 ■
624 1747 !6 
.6663492 iii 
• 588 1673 16 •
■62019910
• 66247 .68l6 •
■681387l«
7162336]6
74b0ii>7;6
•77973386
•311388316
•84'29795 6^
•8745079 6
748897
758965
768998
778996
788960
7P,S890
808786
818618
828476
838272
818036
857766
867464
877130
886766
89636S
906941
916 ISl
924993
934473
943921
963341
962731
972091
931426
090727
OOOOOO
009244
018463
027660
■036811
04.)913
0.66048
064126
•073178
082201
OOni;)
10017(1
109116
•118033
 126926
 136792
1 41634
•163419
162239
•171006
179747
13.J463
197164
20682 1
21446!
22 31 18 i
2;;i6?ii
2 40261
243800
267321
266826
274306
■28076()
291194
299604
307993
316359
,Cubi Rnot
,88 5748807
.'>0,5'7.V<i»(l5
i6iJ67t5rf!W8
,40.677H9!»0
isn 8 •7889(50
,00&'7PSH90
I00l580H786
18H;5'88(U8
i73!58iSl7ti
ltiOj583Hi7'i
lu(iij'Hl8()3i)
100!o8fl77ii(;
04 3b67l<)1
Ib8687713(»
i69i» 88076 "i
!ll5896.'ttis
)01io90i)941
123 1
'ti' .
190 i
198 I
li»5i
188 I
f83i
186 I
199 I
!;u
186 1
471) I
)87
144
Slnl
i'JJ I
JO!) I
JO J I
U)i I
j89 •;•
400 0
309 6'
J4'2 0
402 0
Uft lo
ss.') I ()•
097 •
iM;") 0'
Jiriis
486 !0 •
248 0
331 !8 •
747!0
9l,ilti;i
924993
934473
943921
9..3341
902731
972091
5 98 1 420
090727
fiOOOOOO
6009244
6018403
60>70.'>0
6030811
04.19 13
00.i;)04S
0064120
6073178
6082201
000n<i;)
010017(1
1091 1.>
ll5()3;i
12092.1
136792
141034
lf)341i»
102239
171096
179747
18.i403
1971.'i4
•20.)82 1
■214464
223ii.Si
'2;;i0?ii
C.73l6'2 4i)261
991 6 '248800
76h'0 267324
87 1« 206826
330j6 27430
ii)7 6280i'()(J
338
883
796
079
6291194
6299604
6307993
6310359
8<iUAnE% CITDES, AND ROOTS.
879
No
26a
.Siiiare.
Cube.
Sq. Ibiol.
Cube Root
No,
S<]ii:tri.
Ciil«.
.Sq. Hoot. Ciib» Iliioi
1
0100!)
10194277
Ijn0.)9737
6321704
310 99860
31.66 1496 1 7 7763888 081 128 1
ijl
61610
10387004: 169373770
0323020
317 100489
aiH;V.(H;j 17 tio 11938, B818102
'i<y>
66026
I068l:i7o,1690,(719»
0341320
318 101121
32 1 67132 1 7 • 83 26646 ;0 • 82502 1
■iM
06630
10777216 lO^OOOOOOO
6349001
319 1(11761
32IOI7.69,17^800.67II0^832771
•ii>i
0()049
10974,!93
160312196
63.>7a61
320 102100
32708000, 17 888543810  83990 1
268
0060 1
17173612
160623784
0300096
32l'l03Oll
33076 1 1 ! 1 7  9 1 0472916  H4702 1
269
C70S1
17373979
1609347(^9
0374311
322 1030.) I
33380248; 17 9 113684 41 • 8641 21
200
67600
17676000
161246166
6382604
323^104329
33(;98207 179722008 6801212
201
08121
17779.681
Ifi 16649 14 10390670
324 104970
340 1 2224 1 8 • 0090000 6  808286
202
08044
17 9847 •is
10^1864141
6  398828
325' 106025
3 1328 1 25! H • 0277564 ■ 875344
203
69109
18191447
10 21 72747
640(J96,^
320 100270
34iJ4,M»76ilH055470l 6H82388
201
09090
IS399744
10 "2480768
641 )0i)8
327 ,100.) 29
34966783, 180831 413 6889419
20..
70226
18009626
16 •2788206
0423168
328; 107684
36287562
18 1107703 0890436
200
707.60
18821090
1 6 •SOOoim 0431228
.120 108241
a.iO 11289
18 13.83671 6 903436
267
7128:1
19034103
10 •34013 40
0439277
330! 108900
36937000
13 10.69021 0010423
208
71821
19248832
10 3707066
6447306
33lil09.)01
30204691
1819340610917396
20,1
72301
19106109
16^4012196
6466316
332' 110221
30694308
132208072 0924356
270
72900
19083000
10^.1316707
6403304
333410889
30920037
1824.S287616931301
271
73441
1 990^26 11
104620776
6471274
3341111566
37269704
1827.66669 6933232
272
73984
2012304.3
16 •■4924226
0479224
335 112226
37696375
13 •30.3,)062 6946149
273
74629
20;l40417
166227116
0487164
336; 112890
37933066
13 3303028 !6S.52063
274
7.J076
20670824
165629464
049.)006
337 1 113609
38272763
18 367669316 953943
276
76026
20790876
l6583l>40
05029.60
338,111244
3S014172
183347763 696.6819
270
70170
2102467
166132477 0510830
339:111921
38963219
1841 1962610 972683
277
70729
2l2639;i3
100133170 0518681
3 10 116(300
39304000
184390889I0979632
278
772S1
21484962
166733320 0520619
341110281
39661821
184{;01863;69363f)8
279
77841
21717039
167032931 0534336
342i 110964
40001088
l34rJ32.4200993191
280
78400
219;)2000
16 ■7332006 0542133
343^17649
.10363007
18520269J7000000
281
78901
•221 8804 1
167030640
O64991.i
344:118330
40707684
135472370i7 006796
282
79624
22426708
107928660
6657672
3161119026
410(J3626
185741766i7013579
283
80089
22006187
166220038
6 .6064 16
346! 119716
41121730
1800l07.62702O349
284
80060
22906304
108.522996
8673139
3471120109
41781923
186279360, 7027106
286
81226
23149126
108819430
0580844
348] 121 101
42144192
1 3 •6647.i81 17033860
280
81790
23393060
109116346
0588632
349! 121811 1
42.608649
1868l6417i7040581
287
82369
23639903
109410743
0690202
350 1122600
42876000
187082 6917047298
288
8291.1
23887872
16970.6627
0003864
361 : 123201
43243661
187349940
70.)4004
289
H3t21
211 37609
170000000
0611489
3.621123904
43614>03
1876 10630
7060696
290
84100
24389000
170293804
0619100
363 124609
43980977
187882042
7007370
291
84681
24042171
17 •0687221
6626705
354 '12.6310
41301804
1881.4.8877
7074044
292
8620.4
24807088
170880076
0634287
356l26026
44738376
183114437
7080099
293
86849
26163767
17 1172428
6641852
366:120730
46118010
138079023 7067341
291
80431!
26412184
171464'282
0649399
367 1 127449
46199293
18 •8914130! 7 09.3971
296
87026
26072376
171756640
0060930
3.68128104
45882712
189203379 7100688
•290
87610
2693 1330
172046506
0004444
359 1128881
40203279
1894729.>3'7 107194
297
88209
2019.8073
172336879
0671940
3001129300
4066000.)
18 •9730()00j7 113780
■298
88804
•26103692
17 •2026762
0 079420
301jl30321
47046.831
190000000 7120307
299
89401
26730899
172910105
6686832
362431044
47437928
190202976;7 126936
300
90000
27000000
173206081
6694329
303I 131769
47832147
19062.5.68917133492
301
00601
27270901
173493610
6701769
304!l32496
48228644
190787340,7140037
302
91204
27643008
173781472
6709173
3661133226
4S027126
19^ 104973217 146569
303
91809
27818127
17406:39.62
6716670
360 1339.66
49027890
19 131 1266 '7153090
304
92416
23094^104
17436.69.68 0723961
3i)7 134689
4J4308<;3
19167244117169599
306
9302.6
28372026
174642492 0731316
308' 13.6424
49830032
191333261 7166096
306
D3036
28652610
174.928557 0738006
339,130161
50243409
192093727 7172580
.107
94249
28934443
175214156 074,6997
370,130M0
50063000
1923.63841 17 1790.64
308
94804
29218112
175499288 0763313
371 1137641
51004811
192013603 7185516
309
96431
29503029
175783968;6700014
.372:133384
51478848
19^'2873015 7191960
i>i\)
OuiOU
29791000
17 000810910707899
373;]3912;)
51896117
19 31320797193400
311
96721
30080231
176351921
6775169
374:139876
52313624
19339079017204332
312
97344
30371328
176635217
0782423
375 1.10626
52734375
1936491677211248
313
97969
30604297
176918060
6789601
376141370
53167370
19390719417217652
314
99696
30959144
177200451
6796384
3771142129
63582633
19 41648V8 7 •224046
315
99225
1
31255875
17 7432393
6804092
378] 142384
1
64010152
194422221
7230427
380
8QUAUR«, CUnKB, AND ROOTS.
No.
Square.! Ciilie.
Hq. ItiiDt. Ifiilx Rniit
379 IIMdil
3H0 1 ( t li)0
nH;r I 'ItfttHfl
3H4il<J74.)(i
8«.5i'tW2
3H7
aas
3H!)
SIN)
Uffl
MDTHII
160.')4l
1613JI
IWKIO
16'iHHl
1A3IUM
3H3!l0llJ!)
.194il6.Vj;)i)
3!».5l6«0.i,i
3U6 ir;(i8l(i
397 luTfiOit
31)8 J, 0840 i
I.WJOl
KiUUOO
100801
161004
10J40!i
103210
1640i,i
101830
10i>04!)
160404
107 281
168100
1«8!)21
109744
170.569
171390
]Tiii»\
1730,i0!
173,>i9i
1747Jll
17i")i>0ll
17()4001
1772411
17.S0S4
1789i9
179776
1806a,o
181476
1823J9
183184
184041
18'(900
I8.i7'il
1860M
1871c"
434l8t.:'of
435 iy9:;";V
4.36li90i; >(:!
399
400
401
402
403
104
406
406
407
40H
409
410
411
412
413
414
4i;>
410
417
418
419
420
4'2I
422
423
424
425
420
427
428
429
430
431
432
433
437
438
9
19091
191844
19:!721
4401193600
441 1 194481
M 139939 If)
aH7200O 19
U'i30(i.'Mri9
ftA742908ll9
«61HlM87ilfl
6i;('i2UI04'l9
.'>7(l(;662.">!l9
ft7.>l24J0il9
67900U03;i9
68411072119
(•>88(!.1Hri9;i9
69319000119
<»977617119
60230288 li) ■
600984.>7ll9
01IO'29.Si:i!t
6102(»87<) 19
6J099136!l9
62.)70773il9
630447921 19
63.V2ll9!t'l!)
(il(>000(M)l20
tiU8J01J20
64901808120
■4679213 7
■.»93..S^77
1M922I3 7
VH*2037
A7o:i8.')8l7
.")9.)91797
621116917
6408827 17
67231.')6i7
6977I6OI7
7230829i7
7484 17 7 17
7737199:7
7989899 7
'8242276
•8191332
■874006'i
8997487
' 9248,'i88
94!l9373i7
974984417
000000017
0249S4417
6.'i4.">0827
6.1939264
60430 1 2u
66923116
67419143
67917312
08417929
68921000
69426.)31
09934.128
70444997
709.)794ll20
7147337^'>i20
71991296120
72.51171320
73031032120
73)r;00i)9,20
7408800020
74618461120
7ol.")1448:20
7.5686967120
7622,)024!20
7676.)62iji20
773087 ;G'20
778.54483120
78402752 20
78!)63u89 20
79.507000 20
80002991 '20
H0t>21668'20
^'^^•.'.•37'20
«!.4i. .04 20
;^:.v'j875^j.i
S. :I8 .. _•)
;;:.;>:4o3 20
84027072i20
8460451 9120
■0499377
■0748,')!I9 7
•0997512
•1246118
•1494417
•17 124 JO
■1990099
■2237484
2181.567
2731.349
2977831
3224014
3469899
.3715488! 7
3960781
420o77.9
•44.50483
•4094895
•4939015
■5] 82845
•6420386
•56696387
•5912003 7
•6155281
85184000'20
8676612121
•6397674
•6639783
•6881609
•7123152
•7364414
•760539,5
•7846097
•80865207
• 832()607 17
856653617
•«80iii30!7
•9045450'7
•92844957
•9.523268 7
•9761770i7
•0(J00000i7
236797
2I315(
24950 1
25r;841
2 621 1 17
•268482
274786
•28107
•J873li
•293633
299894
306143
312383
318611
•321829
•331037
•337234
•343420
•349597
•355702
•361918
•368063
•374198
•380322
•386437
•392511
•398636
•4047'2O
•410795
■4168,59
• 42^29 14
•428959
•434991
•441019
•447034
•45;)04(i
•459036
•405022
•470999
•476966
•482921
•488872
494811
500741
506661
•012.571
•518473
•824360
•530248
'536121
•541986
■547842
653688
6595'26
665355
671174
676985
68278i!
588579
594363
600138
605905
611662
No. .'<iiiai»
C'lilta.
142 I9.,:i04
I 13 1962ir
444 1971.3rt
110 I!I8025
116 198916
447 19980»i
1 18 200704'
149 20 100 1 1
150 202.5001
451 203401
152 204804
453 205209
(54 2061 10
l,'5;2fl7025
'"f. Itnot. (iitii. Ulh>'
H63.50<88 2l
86938.107i21
87.V2Ma8J21
HSI2II2,5J
8H7I66.36'2I
8931462321
8991539212 1
90ai8H19'21
9112500021
91 733851 ]Jl
9231510821
l)2959fl77 21
93576664121
0419637521
l5620793tti »48IH8l6i21
157 ,208S J9 9M4399821
l,58.'09764i 9607 i;tl 2121
159 210881
160 211600
101 12 1 2.521
162121.1444
163211369
4641210296
405l2l62^25
90702579i2l
97336000 21
97972181 21
98611128 21
992.52817J21
998!)73 4421
00544625121 •
40612171.56 101 194696^ll •
467 218089 401817663121
I68'219024 102503232i21
1691219961 ! 103161 709I2I
170 1220900
471J22I811
172i2'22784
473J223729
474224676
103823000
104487111
105154048
105823817
106196424
475 225625il07171875
2265761107850170
22752910853I333
22848^41109216.3,52
476
477
478
479
480
481
182
483
229441
230400
231.361
232324
233289
484 123 4256
185J23.5225
186 236496
487237169
109902239 21
110592009 21
111^284641 21
483
189
490
49)
238144
239121
240 hlO
241081
4921242064
11198016s
112678.587
11337990 4
114081125
114791256
11.5,501303
116214272
116930169
11764iH)00
118370771
119095488
493!213049il 19823157
19424403a 120553784
I95I2 45025 12 1 287375 22
496 12460 16
I97247009
4981248004
l')!):2490fll
5UOJ250000
5011251001
502252004
503125.3009
5041264016
1220239361
122763473J22
12350.5992 !22
iM>5M99 22
1 25000000 22
125751601 22
12660600822
127263537 22
128024064
•0'237:W07
•0475652 7
•07i:)075;7
09502317
•I 187 121 7
•I423?457
•166010517
• 1896201 j 7
•2I32034I7
••236760617
••200291 6j7
■28379677
■3072758 7
•3307290,7
■3o4 1.5657
■ 37 7 6.583 j 7
4009346:7
4242853:7
44761067
4709100 7
494185317
61743 487 ■
6406,592 17 ■
.563H.5877^
•587033 lj7 ■
61018287 •
•6333077,7 ■
• 0564078,7 ■
•6;94834 7^
•702.53417
 726,56 lOi 7 •
•74856.I2 7
77 154 11! •
•791494/7
8I74242;7
■8403297; 7
■8632111;"
■8S60636J7
■90,89023 7
•9317122:7
•951498 4'7
97726107
000000017
0227 1557
0»,;4'jV7 7
06S0T6.' ;
0' or .;'!■(
1.00, ,;. .
1359n6 7
1585Zl^S7
18107.K;7
20360J;i 7
2'26110j!7
24.8.59.56 T
••271057 a 7
•29349687
•31.59136 7
■3383079 7
■3606798 7
•3830293 7
•40,53566 7
■4^270615'7
■224499443:7
017412
•623 15 J
■62t<,frl
83100;
•640:i.';
•64602."
•65172/
6.57 I II
663;i9l
•6087(;t;
071431
•68008(
■fi8.57;i;i
691 ;i;.'
69; 00.
7026'J.
70M2;t!i
71381.
71911.'
7250:1;
73(161 I
7361w
741 ?5;;
•747311
•752861
•7.5810:
•763931/
■7694;ij
•7749811
•78019(1
•7859i/:t
•791417
•79fiiJ7l
,802151
80792.)
•813381!
•818816
•82429)
•82il73.)
•83516!'
•810.i!l;.
•34601::
•861421
•856821
86 ■.:■.• 24
•.•'67613
 r,!, 1
37b.k)>
•88373..
•8890!I5
■S9444:
•899792
9051 2i,i
910460
■9i.57,i;i
92110(1
92640,i
0317111
93700..
9122:i:i
947671
952848
963114
M I
, Itnnl. I'litip t(ii,.t
l'J37:W0'7
>i7.)i;.V.' 7
t7i;)i>7.">'7
y.mtr.ui
iia7i2i:7
«itW10j7
!3tl7Hii«l7
!(>0;i<i(lj7
!S37iW7i7
:mi1'M),7
ii.4i.v«<>:7
i77d.WJ)j7
ooiiivirtl;
'Mv'H<3:7
47(>10'i7
70!M(I0 7
W18:>;ji7
1743187'
JOO,S927 ■
f):{HJ877'
87033 1 j7'
10l8i8'7
:«3077;7'
;)tiU)78i7
<"<J183a:7
n:.'.5341l7
4a,J(i3^7
7I54I1I7
IU4M/I7'
l7424v!;7
4U3J97i7
t)3v;illj7
^I)0(>8()j7
W!tO'i3 7
1149817
r;>tii(i7
J000007
W7iAr)!7
ioVji'7 ~,
jsoTfi.. ;•
{59410 7
)8;>zsiiS7
?I07.K7
)3ljOJ;i 7
■10;>7a r
»34!)rt87
o!)13(}7
IBSO/iJj
>06798 7
>30iy3 7
>r)3u(jj 7
!70615'7
.99443:7
GI74IJ
(i.>31,Vj
WH>(5 1
an wo;
HJtJO'i,"
>i}l7ii'
ti,')7ll I
•tirt3;i(i.
•«(w(in
•Ii7l43f.
•«8ll08i;
•fi.J.'u;);!
•tifii3;.'
■OICOI).
•70.>lii,
•7(iM\):i!'
•713'il.
•7I1IJ4.'
•7'i.i0:vj
•73(KJI I
•73t)s,H
■741?.,;;
•747311
•7.'>810.'
•7(j;«);!i/
■7G!i4;ij
■774;)8il
78049(1
78,);t!/3
7SM487
70fiiJ7l
■8()j;,.i
'H070.'.)
81338!!
'8l8Slli
■H.).Vj()
8iil?a.)
83(">lti:'
8 10,"i!l;.
34001::
■8Ji4v!l
•0..68.H
80 ^.74
•f'(i7!i3
■37b.Jo^
•883:3..
•88S<0!i.')
■S9444:
•8907l!>
•i)Ool.*;i
•9104o0
•!)l57.'i.i
ilJllOO
■9i6l0o
•0317111
'.''3700..
■9122:i;i
•947071
•9oi848
963114
SQUARES, cirnra, and roots.
8^1
No.
fiOS
flOO
fl07
ao8
609
010
511
61'.>
813
614
61i>
fll6
,517
518
f,19
620
6Jl
623
t>U
626
626
27
623
o'i')
530
631
632
633
634
635
636
637
638
839
640
641
642
643
644
546
546
547
648
649
650
561
652
663
554
655
6.66
667
658
569
560
<9
602
663
S64
56t)
566
667
Sqimre.
Cube.
fn. lUiiit. ICii'j* llnni
2360'26
2,56(I3<!
2;,704!>:
•26H0OI
26908:1
•jeoioo'
'261 121 1
J6)I41
■!B3lfii
2641 9f)
2B82 .'.)
060266
207 2H9
268324
•2693(>l
■270400
271441
272484
273.'i^29
27467C
276026
•276676
277729
278784
279811
280900
•281961
283024
281089
'286166
286225
287296
288369
289444
290.")21
■291600
292681
293764
291840
296936
297026
298116
299209
300304
.301401
302600
303601
304704
305809
306916
30S02,';
309136
31024!)
311364
312481
313600
314721
3I6c44
316969
318096
319225
320356
321489
r2'i7H7626!29'
1 29.5.112 1 0!'22'
I30:W38J3!'22'
131000612
131872229
13!rt610OO
lft.,430831
l34.l7Vi8
136006697
136798744
186690876
137388096
1.38188413
138091832
I397983.')9
140008000
14I4^20761
M2236648
143066067
1438778^24'22
14 17 031 '20 22
146531676l!22
I46363183I22
I471970.)2!'22
1.1803.")B89 23
148877000
149721291
l/i0.'>6876a
1,M4I9137
I.V2273304
163130376
153990656
1649 VI 153
16.6720872
1 ..06908 19
167464000
168340421
169220088
160103007
160939184
161878626
162771336
103667323
164.366692
16 ■.469149
166370000
167284161
168196608
169112377
170031164 23
1 70.96387 ,".23
17187901023
172808003 23
173741112 23
174ti76879!23
176016000 23
176658481 ■23
177604328 i3
178453547 23
179406144 '23
180362126 !23
181321496 23
182284263 23
472'20m!7'
494443H7'
6160606 7'
6388.V)3'7'
6610 283 7'
683 1 790 7'
60630917
6'274I70 8
6496()33'H'
671 6681 18
09361 U^H
7166334^8
737634018
76961348
7fcl.'>7153
803608618
8'2.54244ia
817319313
86910.13 '8
891046318
912878518
934(189918
9661800 S
97826008
0000000,8
021728!)!'.^
04;14372j8
06612.52'8
086792818
1084400!8
130067018
16107388
1732606 8
1948270 8
21637368
8
8
8
8
3
8
a
8
8
8
3
8
8
8
8
8
8
3
8
8
8
3
8
8
8
8
8
2379001
2594067
230,8936
302:1604
3238076
3462361
3006 129
3380311
4093998
4307490
462078B
4733392
4946302
5109520
5.372046
6684330
6796622
6008474
6^.!>0230
6131808
60.13191
6854386
7006392
7276210
7486842
7697286
7907545
8117618
963374
068027
07.3873
979112
984:114
9H9670
90 17 88
000000
00620.'
01 0403
016696
020779
026067
031129
036293
041461
010003
061748
056886
062018
067143
072262
077374
082480
087679
002672
097769
10^2839
107913
112930
•118041
'12:i096
128146
•133187
■138^j23
14326:1
•1,W270
•15;V294
•1. .8:106
•163310
■I(i8:l0!)
•173302
•173239
•183269
■13,8214
■193213
•19317.'
•203132
•20308
•21302
■217966
222898
227826
■232746
■237661
•242671
•247474
■2623
•257263
•262149
•267029
•271904
•286773
i^jii'ir..
.08 :W2624
.it!9 32.1761
.70 32 1900
•>7I 3^2604l
,.72:127181
673 32H329
071.329476
.■i75 330026
.'i76.331776
,)77 332!»29
678 3310.84
.'.79 336^241
680:336400
68l3.37.66l
682'333721
0831339880
.•.841341066
686 '3 1222,'i
686313306
.,^7l
Ciil*.
H.120Oi;i2
134220001f
18.) 1 03000
186169411
187149248
188132617
189119221
190109376
l91ln297624
192100033,21
19310066224
194101639124
.'<<. itout. Cubt Ktiot
637 344.069
O88I3I.0744
1891346921
090 [3 18 100
')9ll3192.Sl
592I3.0O 164
)93i30l619
691i3,>2836
606 364026
.96
)97
698
3.)5216
366109
367604
6903.68801
300 360000
001
002
003
604
606
606
607
608
009
010
oil
612
613
614
616
616
617
618
619
620
621
622
623
624
361201
302404
363609
:t64816
366026
367236
368449
369664
.370881
372100
373.121
374044
376769
376906
378226'
3794.06
330639
381924
;183161
384400
386641
386334!
:l38129;
389376
026 390626
020
627
628
629
630
3918761
393129
394384
396641
396900
19611 ■2000
190122941
197137:ni8
19816.0287
199176704
■200201625
■2012;)006'i
202202003
•20;^297472
204:136169
•206379000
206426071
207474688
208.027867
209684634
210641876
2117087.36
212776173
213847192
214921799
2IdO0O0O0
217081301
218167'208
219266227
220348364
221416126
•222545016
223648643
224766712
226866629
■226981000
■228099131
229220928
230346397
231476.044
■2.32003375
233744396
■231886113
236029032
■237176069
233328000
239483061
240641343
•241804367
242970624
244i40626i25
24631437626
8327606,8
86372(><ll8
87.10r28;8
8»60U«)3 8
916621 6 ;H
9374134:8
9682971 13
9791.07618
0000000 8
0208243:3
0416306 8
0021188 8
0831, 892 ;s
1039110 8
12467628
1463929 '8
l<i009198
18677328
2074360 8
246491883
247673152
248858189
250047000
2280829
2487113
2693222
2899160
3104916
.3310001
3016913
3721162
3926218
4131112
4335834
4640388
4744766
4948974
6163013
63,)6333
5060683
5764116
5967478
6170673
6373700
6676560
6779264
6981781
7184142
7336338
7.588363
77902.34
7991936
819.3473
8394847
8696958
8797106
8997992
9198716
9399278
9699679
9799920
0000000
0199920
0399681
0699282
0798724
0998008
218036
2,80 l<i:i
291344
296190
301030
306:166
310601
31.0617
3?0330
326147
32996 1
33 17 ..6
3;)fl66l
311341
319120
363906
36 JO? 8
30:U40
:»(i820!>
372967
377719
382466
387200
391942
396073
401?98
406118
410833
416.042
420246
4'24946
429038
■434327
439010
443688
448360
463028
467691
402348
467000
471617
470289
480926
4,86663
490186
494806
499423
504036
508642
613243
517340
622432
627019
531601
696178
640750
045317
649879
604437
663990
66363«
5680S1
6726H>
3S2
SQUARES, C(MJF..«!^ A.VI) ROOTS,
No. 'Sunnre Cuhc.
fi). I!(ml. IcubelLiol
•Ml .'>9<lrtlv'.)l)TO.ifH 2v
•■■',i 3!t;i4Hi,'.V2 i:!5Q(jS i>;)
i;:!:!'J00H8:)ioo;}r.;!6l37'^v
i!:!t 4019.)()'.Vi4,S10104 J.
4P?^'Jol>,G017S7.V2.>
ii;;(> 4(Uloti;.>r)7.',)P4r,(i ir,
;.'<7 40.")7(>:t'.>,s.!74J(,;.TJ j •
:?S 4(170 J4'.>r>0()0407.' '.^r
i:i9 40'3v!l'.>il()<ii7f; J,
)l()40!.it;()0lj()>14IO00.}")
•ill IMissi'o(!;j37172I J.V
i I' 4 1 ■.' 1 04 : OH JfiODJsS in ■
ilT4i:i!i;>>(!r)847707Jo
■:il 4n7:i(i'.Hi708098i.'o
(i lo 4 1 <nii;)!JK833()l ^rlJ j ■
;'l(i 417316 ■,>60:J6Hl;5(5i)r)
il7 ll^f)0!l:708400jn iV
■>tH4l!i;ii)4 ■>720!)779''J.V
iil!)421J01 •J733:.914P','V
.i;!0'4','^::)0()J74i;.>.,000:2o'
ti:)l.'4."3:!0l l.>7J8!»44;il v'o'
:v..' l>')l()l!J77lti?8()S'0v
'i"i3 l•J(jn9■>7844.^O77:■i•V
i^Jl:lJ7710 07f»7•2rl,>^;^\•.j•
tJ•i.VlJr)()•^,•J81!)I13^;Vl>,j•
ti)'):4n033'j!J8.>3n04!fi •>^>•
^K■>7il.■^l^)l!)■2S3.)!l33!13,•2•')•
l>•j8'■^■.•^'{ji •2848903 1 ■_> i.v
(>")!•, 13428 1 !'28(i I !tll 79 20 •
i)tjOi43.')ti()0i28740HOOl) 25 •
ot)l43fJ!l:2l{;SS8fl478l'0o
t)G2!43824l 2r!0117iV28
1197131:8
l3!Hil(i2i8
K"'9l!)13 8
I793.jf)i; 8
l9r)20(;3;s
'2l!)0!0l!8
•238^08918
•2")3tjf)lfl!8
'2784H>3ls
•29822 1 3VS
3179778,8
3377189'8
•3)71 147i8
■377ir),")l i8
■3;'()8">(VJ8
■41H.)3()li8
4361947IH
4r!5844]J8
47^4784:8
49.")i)97()tS
yii7i)U)l8
o342907!8
00389 1718
•5734^3718 •
(3t)3j4.'.':)o69!29M3t2l7l2o
■■if)i:41089(v2927r)19942.)
■■iH)!44222o294()79tJ2o J/i
^5iiti:4 13V")()!29o4()829(i'2.')
H!J7'414.i89l2!Hi7l(t9h'3 20
■>;i8;Mir2J4!2;t8()77fi3'2:i.)
■»i;ru7:,(il!jl9,U830r) 2;)
t)HVI48!mi)'3.')..)7()3imo;2")
ti7iU.'!()2ii3!)2ill7nlv!.">
■ i;2;l=)l:)843()3tf!l448:2.>
()73'4V292!l]304S21217>2i)
(iri'4o4 27tij:)()dl82024'2;)
(>r:)ilo.")li2.)!307")4t;87")!2;)
i."li;i ,(i97t);30Sr)I577(i!2l5
ii77i4^'.:832i>':'l(V288733 2(5
iir8!4.yr;r?i:31l'o(.;.)7o2 •2()
ri79!4ol04i:3l3i)4(i8392H
(iUl!!)2400;3l4l3'2(lOO2«
<814f;37«:3ir)82I24i;2t)
ii824ii..IMl3172l4.')6826
oM3!4:ji;i8;i :;!^iJ!I987'2(i'
o84'4(>?8.5(V32(!OI3i()l'2G'
i86lc)02213:!4iyi2:V2b'
;.«470.=)9(i 3:'28288'ji)'2()'
)87!!719S9:3242;27(!3J()'
i88'4733Hv)2.>(iijM07>!'i)'
•i89;47472r32("n827(J9i26'
'i'JU,17(il(IO328J0!)OO()'>Ci
li:) 1:4771^/1 32:)93937li2fi
l)92:478S«4 33I373888'2ti
093 4 802 19 3323 1 2057 j2S
o929(i,8ia
t)l2 190918
(i320112j8
<i.">l 0107:8
67099o3'8
090 ltJ.V2J8
'7099203'8
7203t)07!8
74878G48
7i)8197ol8
78/093918
8{)(i97o8:8
82(i343li6
840tj9(i0.8
8t)ot!313.8
•S.il.'ioSJid'
■903iit)77l8'
•9229!i28i8'
• 942243 jl8'
•9()iol(ia8'
•9807<)2!3
•uooouooL8
•01 9223718 ■
• 038433 iiS •
•0o7(v28t'8
•0?ti80;it;'8
•09o97ti7'8
■llol2978
•1342fH7l8
■lo33937i8
■172o(!47 8
•191(i017'8
•■2101)848:8
■:2297oU'8
•0488090 Is
■2li78.)li;8
•281)8789. 8
30o8929'8
3v>48ii32'8
0771.V2
o81(Hl
'.■)8ti20o
090724
o9o238
099747
U0^2V2
6(18703
t)l32l8
617739
•(werou
•031 183
U3oUt5".
(i 101 23
(i44o8,)
(J4904 1
003497
607940
oe23!' I
00683 1
67 1200
070697
6801 21
(is 1046
68,3903
•(;9337()
■69778 1
702188
706087
710983
7I037;1
71970!i
724141
7280 18
73289'>
737260
741021
7 1098.;
700340
704imi
709;i3n
703381
707719
Ii9l 4S1630
Ii90 483020
Mi 481 no
(i!'7 480809
:i98 487204
0<)9 488601 1
700 4900()i)i
70 1.491 401!
702 492804
703 4;)4209
704 4!t0616
700 497020
700 49843(i
707 499819
708 OOlOOl
09 0O26'ii:
lOOOUOO^
711 000021'
712 006944
713 008309
714 009790
710,011220
6 012000
17:014089
18,010021
719 01090!
720 0184IJO
721 019841
722 02 1 284
723 022729
1/V.'4i76
720'0'.'o()2o
720'.V27076
7O7:O28029
728:o;2998 1
72!l:0314 11:
7 30.);!O900
731 !03 1361
732:030824
733:037289
3 1! 038700
730:040220
73:i'04I()96
737i043l09
33 1200384 '26
:330702370 20
337103030 2^
j3:!80(;8873 20
31006.'I392 2»)
31103209920
:J 13000000*20
!34U72l01'26
3409 18408 20
1 34 7428927 06
3489i36i)4'20'
1300402620 20
301890816 ■26
;303393243:2()
3048!1492'20
:300100829 26
;307yll000'o(j
'309420l3l:2o
36094412826
302167097 26 
i3039fM3ir20
!360020S70:o(j
'307001096 •20
30860 181 326 •
370140232 •20
371094909:o{j
37 3248000 20
;37480030li26
370307O18'J6
;377933067:26
379003121 26
381 078 1 20, 26
382007176 20
38 124008326
3:J0^t2o30>:20
387420489 27
3890!7000'27
SOO'j 17891127
•3^38797{8
•3628027 8
38181 ni's
•400707 iLs
•419fi890J8
• 4380081 18
•40701 3 1 Is
•4764046!8
•490'2826jri
•0141472iS
•53299:«:S'
0018361 18 •
■0700.i00!8'
■ 08947 10 '8 ■
6082694 '8'
02700398^
640,S202l8'
6(j 10833 18^
683328) ,8 •
702009818
7^20778 1,8
739183918
70817031b
(9002
:3922'231 0827
393832837'27
390440904:27
397O00370!27
39y088206i27
400315053,27
7700o37ri8;0446I4 401947272 2;
770383
■78070,:
• 780029
•78934
•793609
•797908
•8;>2'272
■800072
•8!()8(i^
•810100
■81MJ7
■823731
828009
83228)
8300. )0
8 10823
810080
849344
739:0 !01 21
740 047090
7il 1019081
7 1^2 00000 1
743:002019
744'003030
7.10:000020
710 006016
747;Oo8()09
748:009.001
74 9 'OO 1001
0' 062000
701:061001
OO.JOOt
703:067009
704068016
7oOn700JO
70'jjy7iu30
1 I
l03.)8341927
,49022 1000''27
406.869021 ;27
4080184 88'27
41 01 70407 '27
4n83078l'07
413493620:07
410100936 07
4J6830703;27
41800899027'
42O18Q74907
401870000^27'
4030C47Oli27'
40000900S 07
400907777i^27
408001 064^07
430368870107 •
,43^2O8iO16:07
■20j8
•81417.048
•83281 )7!8
•80141 iOJ8
•870007 7 18
•8886093 8
•9070481 J8
•9008040:8
•914387018
9609370J8
9811701 8
•0000000 9'
•OI80100!9'
•03701 17';)'
•000498019 ■
•07397079^
■()901314!9^
■11(»'^834'9^
•1093199i9^
•1477439J9
' 166 1004; 9 •
1840.044 9 •
0009 11019 
0OI31O0I9
0396769,9
■•208006319
•0763034'9
•09 16881 !9
•31'i0000j9
•3313007 9
•34908879
367 8644(9
•3861079:9
•4043790i9
•4'206184'9
■4408100y
■4090!;04'9
'4772633in
■8078.lb
'860090
'866337
870.07(7
874310
879010
HS3366
•887488
•891706
•89.0900
•900130
•004336
■90803.8
•91 2737
■916931
•901101
•900308
•909190
'933608
937843
940014
946181
900311
601003
908008
960809
966907
971101
•970010
•970376
• 933009
■9b7f;37
•991760
•9908S3
■ 000000
■004113
■ 008203
■010309
■016131
•000009
•001004
•028710
•030800
■036886
■040960
040041
■049114
003183
057048
061310
060307
•069400
•073473
■0770'2O
■081063
'080(>03
089639
093672
097701
101726
10074s
109766
777 (
778 (
779 (
780 (
781 t
780 (
783 (
784 I
780
786 (
787 1
788 1
789 (
790 1
791
790 1
793 (
794
79C
7961
7971
7981
799 (
800 I
801
800 I
803 1
804 (
805 1
806 (
807
808 (
8091
810 (
811
810 (
813 1
1 (i
817 (
818 1
819 (
Sq. Hoot.
Cube
Hoot
t!a«.'8:V27rt
ti3Siyini;8
u4007i'i7 iLs
(i.tlilfiSfitjis
IV ').)".)) 3 18
l)'17640K>!8
i)40iVi8.'6jri
lJ51414;.>iS
vn3..!f);):«jS
"■0r>l836ll«
J<>70C.>();)!S
)oSi)4710'8
)608J(ili4'S'
rti()t08.'j38
i70.'()/iit,i.S
r7,'U77.Sl,8
i73SUs:;!i;8
i7:)8l7u:i!8
i" ?7i)y.>;'irs
i7it).Tj:2(tl8
i81417o4.8
i8:{osl)7!8
•8.)144iJJ8
■8700;)77'8
•88S60(t;jU
•!>II7'.'4818
•(ii;,SJ40:8'
•(Ul.'5s;Ji8'
•!)ti:!);!7j]8'
•!)sli7;,ii3.
•dOODOUO!)'
•0:1701 17';) ■
•();■).")4!^).,;)■
•07:!!t7J79
■()f).>l:!J4!0
•lHCi834!fl
•1:!!!310'.)!()
•1477439';)
■ItiUlo.Ujf)
•I84.).>!4;)
■iO'OllO.iJ
•:i.!l31o2ii)
■i;l%76y 9
■•'f>S0itJ3 i)
■■>7f)3'j34 ))
'v'yi<>s8ii9
SridOOfij!)
3313007!)
31l),)8879
3078044(9
3S61xV!'i9
40437;)oJ9
4'22iil84i9
44()8».;);V<)
4090004^9 ■
477J033i9'
49644>2!)'
'863.')98
8u784b
86i.'09;)
866337
870070
874310
•879040
•8S330H
•887488
•891700
•89.59a0
•900130
•004336
■908538
•91 '2737
■910!)31
•921 lJl
•925308
•9v'9U)0
'933008
937843
942014
•946181
•950344
■054503
•958058
•9028U9
•900957
•971101
• 975240
•979370
• 983509
■987037
•991762
■995HS3
' 000000
004113
0082'23
0123'29
01 643 1
• 021)529
•024024
•028715
■03280^2
•030880
•040905
•045041
•049114
•033183
•057248
•061 310
•005307
•069422
•073473
•077520
•081003
•085003
•089039
•093672
■097701
101720
10574s
109706
SQUARE.*, CUIJES, A.V5 ROOTS.
383
Nn, Square. I Cube.
7.57 573049
758 574.564
7591576081
577600
579121
580644
582169
583696
433798093
43551951'2
437246479
438976000
440711081
P(). Rool.
•276136.330
:27 6317998
275499546
276680975
275862284
Cube Uotii
442450728 276043475
444194947
 445943744
580225 447697125
586756 449455096
586289 451217663
589824
591301
592900
594441
595984
597.529
774 599076
775 600625
776
002176
603729
605W4
606841
008400
609961
611524
013089
452984832
454756609
456533000
45. ■^14011
460(»i, >648
401839917
463684824
465484375
467288576
469097433
470910952
472729139
474552000
476379541
478211768
48004808
276224546
276405499
■276586334
276767050
276947648
277128129
117793
121801
125806
9129806
9133803
9137797
9141788
9146774
9149757
9163737
9167714
No.iSqn.ire.l Cube
9113781 820 67240e65I 368000
"•' ■■'^" 821 1674041 155.3387661
Sq, Rool. {Cube Root
4855876") 28 035691 5
4874434i»3
614656 48 1S90.3O4
616225 4837366*i
017796
619309
020944
622521
624100
625681
627264
628849
630436
032025
633616
277308492 9161606
•i7 7488739 9165656
277668868
277848880
278028776
278208555
278388218
278.567766
278747197
278926514 .,
279105716 9
9169622
9173585
9177644
9181500
9185453
189402
193347
197289
:01229
27.Q284301 9205164
279463772
279642629
279821372
280000000
280178516
797 635209
" 3 036804
) 038401
) 040000
1 6U601
! 043204
I 644809
046416
I 648025
' 649636
051249
662864
054481
656100
657721
659344
813 600909
'^14 6«2596
31 1664225
816,6058.56
817
489303872
491109069
493039000
494913671
49679'>'^88
498077 io7
500566184
230535^203
•23071.3.377
280,891438
28 J 069380
28 1247^222
•281424940
231602557
231780056
502459375 281957444
504358330 282134720
50626157.3 282311884
508169592 282188938
510082399 232665881
512(JD0000 282842,712
513922401 283019.434
515849008 283196045
517781027 283372546
818
819
019718464 283548938
021660125 283725219
023006616283901391
520557943 284077454
527514112 284253408
529475129 284429253
531441000 '284004989
533411731284780617
535387328 284956137
537367797 285131549
539353144 286306852
541343376 286482048
o4333«4»6 236657137 i,<ytvioi
667489 54533851 3128 • 58321 1 919  343473
_. ...,.,,.. ._..,..,^u3:tj;? ja;;»BD
670.oi549353369J286181760 98»«O95
9209096
9213026
9215950
92^20873
224791
228707
23^2619
23752S
240433
244330
2482.34
252130
256022
259911
!)• 263797
9267680
9271559
9275435
9279303
928317S
9287044
9290.907
•294767
•298624
302477
306328
810175
314019
9317860
9321697
9325532
9329363
9333192
9337017
9340838
9344667
822 675684
823677329
824678976
825J680625
820i682276
8271683929
828 1 685634
829 i 687241
8.30 688900
831 1690061
8326922i24
833!693889
334690056
830 697220
830693390
8371700569
28
550412218
657441767
659476224
561515626
563559976
665609283
067663553
569722789
.571787000
673806191
576930308
578009637
680093704
582182876
584277056
 586376203
838 702244 588480472
839 703921 590589719
840 706600 592704000
841 707281 594823321
842 708964 590947688
813 710649,599077107
844 712330 601211584 29
845 714025J603351126 29
846715716j605495736 29
8'*7 717409 607645423 29
848 719104009800192 29'
3 19720801 01 1960049 29'
8o0 722500 014125000 29
"" 29
24201
725904
28
2.3
23
28
•m^
•28'
•28'
28
28
28
28
28
•28
23'
28'
'28'
28
28
28
28
'29
'29'
29
9
9
9
9
9
9
9
9
9
9';
9^
9^
9
9^
016295061 1.^.,
618470208129
727609 620000477
729316 022835864
731025 626026376
732736027222016
734449629422793
736164 631628712
737831 033839779
739600,036056000
741321038277381
743044 640003928
744769
746496
865748225
042735647
044972644
647214626
866 749900 049461896
"^^^.^
807 751689 651714363
868 753424 663972032
869 755161 656234909
370756e0O 65350.3000
758641 66077631 1
760384 663054848
762129 665338617
63376 667627624
65626 669921876
707370672221376
709129 674520133
770884 6768.36162
879)772641679141439^5,
880)774400j631472000J29
Cioii770ioli0837y?»fjj J29
882^777924686128968 29
871
872
873
874
875
876
877
878
9
29
29
29
29
29
29
'29'
■29
29
29
29
29
29
•29
•29 •
29'
29
•29
'29
29
29
29
29
29
29'
29
■6356121 !9
6530970 9
6705424
•6879766
•7064002
7228132
7402157
•7670077
•7749891 J9
■7923601 9
■8097206
■8270706
8444102
•8617394
8790682
•8963606
■9236640
9309023
9482297
9654907
9827635
0000000
■0172363
■0344623 _
■0516781 9
•0688837 9
0860791 9
1032644 9
12043969
•1376046 9
•1647595 9
•1719043
•1890.390
•2061637
■2232784
■2.103830
■2574777
274.5623
2916370
•3037018
•3257666
•34'28015
•3698.365
■3768616
■3933769
■4108323
4278779
4148637
4018.397
4788059
•4957624
■5127091
6296461
6466734
5634910
•5803989
•6972978
•6141858
■6310648 „
■647932519
66479391a
681644219
69848489
359902
303705
367005
371302
37.5096
378837
382676
380460
390212
394020
397790
401509
405339
409106
412809
410030
420387
424112
427894
431642
435333
439131
442870
440007
450,341
461072
457800
401520
465247
463906
472632
476396
480106
483813
4876181
491220
494919
498616
602308
•605998
■609686
■513370
517051
520730
524406
528079
531749
•535417
■539082
5427.44
546403
550059
663712
•507363
■561011
•664656
608298
671938
675574
679208.
''5S2SdQl
686468 {
6900941
1
9
9^
9
9^
9
9
9^
0
9
9
9
9
9
9
9
9
9
9
9
SQT>fcRES, CUBES, AND ROOTS.
\o.
Scjuare.
Cuba
't3 7796SS1 G88.l6o387
>84 781466 0f)0807I04
85 78332.5 15<)3151125
30 784990 6ft56064.56
o87 786769 697804103
;i68 788344 700327072
189 790321 702595369
890 792100 704069000
191 793881 707347971
892 796604 709732288
893 797449 712121957
394 799236 714516984
395 801025 716917375
896 802816 719323136
897 804609 721734273
898 806404 724150792
899 808201 726572699
900 810000 729000000
901 811801 731431701
902 813604 733870808
903 815409 736314327
904 817216 738763264
905 819025 741217625
906 820836 74.3677416
907 822649 746142643
908 824464 748613312
909 826281 751089429
910 828100 753571000
91 1 829921 756058031
912 831744 768550528
913 833569 76104849
914 835396 763561944
916 837225 766060875
916 839050 788375296
917 840389 771096213
918 842724 773620633
919 844661 776151559
920 846400 778663000
921 3^8241 781229961
922 S50084 783777443
923i851929 786330407
Si}. Hoot.
29
29
29
29
29
29
29
29
29
29
29'
29
29'
29
29'
29
29
30
30
SO
SO
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30'
30'
30'
30'
SO
SO
30
30
30
■7163159
■7321375
•7489496
■7657521
•7825452 9
■79932S!
8161030 9
Cuba Root
8328678
•8496231
■8663690
■8831066
•8998328
'9105506
9332591
9499583
9666481
983.3287
•0000000
■W 66620
•0333148
•0499584
•0666928
•0832179
•0998339
•1164407
■1330383
•1490269
•166206319
•182776519
■1993377 9
■2158899 9
■2.3243299
■248966919
■26.5491919
2820079] 9
■29851489
31501239
•33150189
' 347981 :j 9
364452&,9
380915119
3973683 9
92 1853776 788889024
935 855625]791453126 30^4138127 9
9J6 857476;794022776 30^430'3481 9
937 859339 798597933 304466747 9
933801184 799178752 304630924 9
939:863041 801765089 .304796013 9
930804900 804357000 30^4959014 9
93l86676180695449l!30^6122936'9
932i868634;S09557568;3052867509
933jS704b9.3121663.'i7j305450487 9
931872350814780504'30C614136 9
935 874235;8174O0375i306777697 9
936:876096, 83003685630 • 6941 1719
937877S69,S33656953 30^6104657 9
938;879844 635293672130 • 6307857 9
939.t;8172l837936019j306431069'9
'140 883600:830584000 306594194'9
41 885481.83.3237621 306757233'9
593716
•597337
•600956
•604570
•608182
•611791
•015398
■619003
•623603
•626201
•629797
633390
■636981
■640569
■644164
647737
■651317
'654894
6.58468
663040
665609
669176
672740
676302
679860
683416
086970
■090.521
•69406!'
•697615
•701158
•704699
■708237
•711772
•715305
■718S35
■722363
•725888
•729411
■732931
■736448
■739963
•743476
•746986
•750493
•753998
•767500
'761000
'764497
•767993
771434
'774974
778462
'782946
785429
788909
79238e
796861
799334
Wo.
S<uaie
942
943
944
946
946
947
948
949
950
961
962
963
954
985
956
967
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
079
930
981
982
983
984
986
986
987
988
989
990
991
992
993
994
996
996
997
998
999
1000
887364
889249
891130
893025
894916
896809
898704
900601
902600
904401
906304
908209
910116
912025
913936
915849
917764
919681
921600
923521
925444
927369
929296
931225
933156
935089
937024
933961
940900
943841
944784
946729
948676
950628
952576
954529
966484
958441
960400
962301
964324
966289
968256
970225
972196
974169
976144
970121
980100
982081
984064
986049
988030
990025
993016
994009
996604
998001
1000000
Cubs.
Sij. Root.
Ciihe Koot
636896888
838501807
841232384
84.3908625
846590536
849278123
861971392
854670349
857375000
860085361
862801408
86.5523177
868250664
870983875
873722816
876467493
879217912
881974079
884736000
887603681
890277128
693056347
895841344
89363212.=
901428696
904231063
907039232
909863i*i<9
912673(».0
915493(),1
918330C48
921167317
924010424
926869376
939714176
932574833
935441362
938313739
941192000
944076141
940966168
949862087
952763904
956671626
958636256
961504803
964430272
967361669
970299000
973342271
970191488
979146657
982107784
985074875
938047936131
991026973131
994011992131
997002999 31
100000000031
30
30
30
30
30
30
SO
SO
30
SO
SO
SO
SO
30
30
•St
30
30
30
31
31
31
31
31
31
31
31'
31'
31'
31'
31 ■
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31
31'
31'
•6920185
•7083051
'7245830
■7408523
7571130
•7733651
•7896086
•805ai36
■ 8320700
•8382879
•8544972
•8706981
■8368904
•9030743
■'■ 193497
9354166
•9515761
■9677251
•9838663
0000000
0161248
0322413
0483494
0644491
0805405
0966236
1126984
•1287648
•1448230
■1608729
•1769145
•19^39479
•2089731
•2249900
•2409987
•2669992
•2739916
•2889787
•3049517
•3209195
•3368792
•3528308
■3687743
•3847097
•4006.309
•4165561
•4324673
•4483704
■4643664
■4801525
■4960315
•6119025
•6377665
■5436206
5594677
'.>r.5.3068
6911380
6069613
6227766
9
9
9
9
9
9
9
9
9
9
9
9
9'
9
9
9^
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9^
9^
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9'
9'
9
9
9
9
10
•8O3804
'806'r/l
80973O
813199
•816655
•820117
•833572
•837035
•830476
•833934
•837S69
•840313
■844234
•847692
■851128
■854562
857993
•861432
•864848
•868373
•871694
•876113
•878530
•881945
•885367
•888767
•892175
■895580
■898983
■902333
•906782
•909178
912671
916962
919351
922738
926122
•939504
•932884
•936261
•939636
•943009
•946380
•949748
•9.531141
•956477
•959839
•963198
9665,55
•969909
■973363
■976U12
•979960
983305
9866 H)
989990
993339
9%'666
000000
Nu. (iij
\iy.
1
2
3
4
5
6
7
8
9
10
11
13
13
14
1,5
l(i
17
19
30
21
33
T.i
•Ji
I
Sij. Root.
Ciihe Koot
M 6920 185
307083051
307245830
307408523
307571130
307733651
307896086
308058136
308320700
308382879
308544972
308706981
308868904
309030743
30 •'•192497
3* 9354166
309515751
309677251
309838668
310000000
310161248
310322413
310483494
310644491
310805405
310966236
311126984
311287648
311448230
311608729
311769145
31 19^9479
312089731
312249900
312409987
312569992
312729915
U 2889767
11 3049517
(13209195
113368792
113528308
113687743
113847097
(14006309
(14165561
H 4324673
il 4483704
H 4642664
il 4801525
1 4960315
16119025
16277665
15436206
15694677
1 .1753068
16911380
16069613
16227766
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9'
9
9'
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9'
9'
9
9
9
9
10
•8O'.'M04
 8062V 1
809736
813199
8I66.)5
82011;
•823072
•827025
•830476
■833924
•837369
•840813
•844254
847692
851128
854562
857993
86 1422
864S48
•868272
871094
875U;}
•878630
•881946
•885367
•888767
•892175
895680
•898983
■902383
■906782
■909178
912671
916962
919351
922738
926122
•929504
•932884
•936261
•939636
•943009
•946380
•949748
•963114
•966477
•959839
963198
966555
■969909
•973262
■976012
■979900
983305
986649
989990
993329
996666
000000 i
TABLES.
3S'>
•^'^^ ^^ ^^ T"^^ AMOU NTS OF £1 AT COMPOUND INTEREST.
3 per cent
Mo. of
I'ay
mcnt]
4 per cent
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
26
per cent
103000
100090
109273
!• 1255 1
1169'27
119405
1 22987
126677
1 30477
P343!42
1 •38423
1 •42576
1 •4U853
1 •51259
r 65797
1^60471
1 •65285
70243
75351
806 11
86029
91 610
197359
203279
209378
l:
1
1^
1
1
1
!•
1
1
1
1
1
1
1
1
2
2
2
o.
04000
08100
12486
16986
21063
26532
81593
36S5T
42.331
48024
53945
■60103
06507
73168
80094
87298
94790
02582
10685
19112
27877
23o!Jri2
240472
266330
260584
1 •05000
!• 10250
1^ 16762
1 •21551
1^276^28
!• 34010
1^40710
147745
1^56133
1 •62889;
I • 71 034
179586
188565
197993
207893
218287
229202
240662
2 •52695
265330
278596
292520
207152
322510
338635
6 per cent
•06000
•12360
•19102
26248
■33823
•41852
■50363
59385
68948
Vo. of
Pay
meiita
179086
89830
•01220
•13293
•26090
•39656
■54035
•69277
85431
02560
20713
39956
360354
381975
401893
429187
20
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
43
49
60
3 per cent
216659
222129
228793
235657
242726
250008
2 '57508
205233
273190
281386
289828
298523
307478
316703
326204
335990
346070
•56452
67145
78160
89504
01 190
13225
25622
■i per cent
438391
77247
88337
99870
11865
24340
37313
3 •50806
364833
379432
394609
10393
26809
43881
616371
80102
99306
19278
40049
561651
684118
607482
631782
657053
683335
710663
5 per c«nt
6
5
6
6
6
7
!■
7
8
8
8
9
9
10
10
11
56567
73346
9'2(ll"
11614
32191
53804
■76494
•00319
25335
51601
79182
•08141
•38548
•70476
•03999
•39199
•76159
•14967
•5671.'
•98,501
43426
90597
40127
92183
46740
6 per cpiil
6
6
7
7
8
8
9
9
10
10
11^
12
12
13
14
16
16
17
18
■64933
82235
] 1 11)9
41839
74319
•08.^10
•453;t9
•810:>LI
■25 1(1 2
'6StUtJ
I 1725
fi3(l(rj
15125
70351
•28572
■90280
•55703
•'23045
■9.3543
■76401
■59049
•46592
39387
37750
42015
J'ABLE OK THE AMOUNTS OF AN ANNUITY OF
£1.
Su. (ilj 1
P.iy. Spercenl. <pcrci;iit operctfiit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
l(i
17
IS
19
20
21
22
•23
•2i
I
6
7
8
10
11
12
14
15'
17
18
20
21
•23
25
:i)
28
)0
!2'
il
oonoo
•03000
■09l)i)0
J 8363
30913
•40841
• 66246
•89231
■15911
■463t*S
80779
19203
61779
08632
59H!)!
!M!8!^1
7(il.i'.tl23
'iU13l25
lH>'i7J27
•"'7(!37 29
OTorsUi
53{i7Sj34
4.V2s,ii3(i
46l7J3:l
4^y20'41
1
2
3
4
6
6
7'
9'
10
li
13
15
16
13
20
00000
■((4000
12I60i
•■24610
41032
63297
■89829
•21423
58279
00611
4vS63;
1
6 per eentl'
00000
05000
15250 3
31012 4
62563 5
80191 6
14201 8
954911 9
1102656 11
1257739 13
M 20679 14
02580 1591713 16
626Sll7^7i29!i 18
29191 19*?)9S63 21 •
0235921 •o7S.i6 23 •
3 per cent
■« per cent 6 per cent
•S2J.03''2365749
■69751 2584037
■61541.28I323S
67123 0053900
778(»,M 33 •06595
96920 35 •710.).:,
■2171)7 ,{850521
0I7.'D41.130!/
(lo'iiid ■I4^5(12U0
25 (
28.
30f
337
367
39  P
43 S
fti;i
.50 ■ S
6loWl 4V,;.'71tli54
50
1866304
SO 70963
293092
521885
757641
000268
250276
507784
7^73018
»• 46208
327594
317422
M5945
223423
>40126
?• 66330
!02320[l04
'•48389 110
I048U 115
!719y6 121
■•50146'l26^
• 396.50 '132 
• !0.S39,139
•5106,VI5
■796tt7jl5'2
44
47
49
52
66
69
62
66
69
73 ■
77 •
81^
85
90
95
99
•31174
•08421
•96758
•966'29
•08494
•32833
■7QU7
■20953
85791
65222
59831
70225
61
64
68
62
66
70
75
80
85'
90'
95
101
•11345
•60913
•4025S
•32271
•43886
•76079
•29329
•06377
■0669t;
32031
8363i'
628 M
6 percent
o9
63
68
73
79
81
90
97'
104
111
119
J 27
97034 107 •70954 135
•09502 145 •
•79977 154 •
•8397(;i65^
'•23175 175
■99334 187
114
120
127
•4091
■02551
■82654
■31960l35
01238 142
412881151
02939J159
87057 168
94539 178
■2632il88'
53.';73:li)S
667081 209
I
14300
•7001(..
■68511
■11942
025.3'>
42(iGu
a47Pii
199
212
22(i
211
256
■?2
00
•15638
•70576
•62311
■03980
■05819
■S0168
88978
•34316
•18375
•43478
•12087
•26812
■90420
•05846
•76196
04768
95054
•6075e
■76303
•74351 j
■50812
09861
56453
95840
33590
Ml
III
386
TABLES.
TADLR OF THE PRKSKNT VAM/F'.S OV AN AN'XITITV
Vo. of
IMy
I
3
a
4
6
6
7
8
9
10
11
li
13
14
IS
16
17
18
19
20
21
22
23
34
33
OF £1.
3l'erc«iit j<i,trcBm
f' larceiitl i)»r cant
0970&7
l!)i;)ir
28:28(il
37J71()
4Wt)Tl
a41719
7 01 oaf)
7 78611
8 53030
995400
1063490
U 29607
U 93794
1266110
13 16612
1376351
14323S0
148774,!
15 •41602
1593692
1644361
1693664
1741316
oaoisi
l8ti6l9
277519
36^999
44dl)^i
5a4214
600.>0o
fi 73274
743533
81J089
876058
938507
998565
10 56312
ll11849fl0
1165239 10
1210567
1265940
1313394
1359032
1402916
1445111
1485684
1524696
15 •62208
95238
85941
75326
54595
32948
07569
•78G37
•46321
• 10782
•72173
30641
86325
3SI367
89864
094310
183339
267301
346510
4 21 236
491732
658238
630979
680169
736009
7886a7
838394
8&V268
929498
N'o. nt
l>ny
■i per cent
•< per car.'.
37965 971225
8377711010589
27406
68968
•08533
•46221
•82115
10300
48357
79864
09394
1047726
1082760
1115811
1146993
1170407
1204158
1330338
1265036
1278335
26
37
28
39
30
31
32
33
31
35
36
37
38
39
40
41
43
43
44
45
46
47
48
49
60
17
IS
18
19
19
30
•20
30'
21
31
31
33
33
33
33
33
33
33
34'
34'
'24
35
35
35
25
•87684
32703
76411
18346
•60044
00043
38S77
•76579
13184
•48722
■83225
•16724
49246
80822
11477
41240
70136
5 per cent I 6percfu»
lt> 98277
1632968
16^6306
1698371
1729303
1708849
1787356
1814704
1841119
13 •66461
IS •90828
19 •I 4258
1936786
1968448
1979277
1999305
2018563
931902037079
3064884
3072004
2088465
2104393
2119513
3134147
3148318
35428
51871
•77545
■ 02471
•36671
60166
73977
1437518
1464303
1489813
1514107
1637245
1559381
1580367
1600255
1619290 14
163741S 14
1664685
1671138
1686739
1701704
1715908
1729436
1742320
1764591
1766377
1777407
1788006
1793101
1807715
1816873
1835593
14
14
14
14
15
15
16
15
16
15
16
15
16
16
15'
I
•00316
■31083
40616
69073
76483
93908
08404
33033
•36814
•49824
62099
•73678
•84603
■94907
■04630
13801
32454
30617
33318
46583
62437
68903
66002
70767
76186
IRISH CONVERTED INTO STATUTE ACRES.
Iriah.
SRltUU.
R. P.
u
10
30
1
2
3
A. n
1
3
p.
1
3
4
6
8
16
32
r.
n
26
Ml
3
6
13
Iriih.
34 24
9 172
1 34 ll
A.
1
2
3
4
6
7
S
9
10
Statute.
Iriih.
A.
K.
p.
T.
A.
1
3
19
6
20
3
38
10
30
4
3
17
153
40
6
1
36
21
50
8
15
26.1
ij
6j
111
100
9
3
35
200
11
1
14
300
12
3
33
400
14
2
12
17
600
16
31
22i
1000
Statute.
A. B. F, T
32 1 23 14}
48 2 16 6i
64 3 6 28/
80 3 38 20l
161 3 37 101
323 3 34 2l
485 3 32 2
647 3 29 13
809 3 26 23
1619 3 13 163
VALUE OF FOREIGN MONEY IN BRITISH,
Silver being 5*. per ounce
1 Florin is worth
16 Schilliiig.s (Hamburg)
1 Mark (Frankfort) .
1 Franc
1 Milree (Lisbon)
8 Reali . .
». d.
1 8
15
9i
4 8
3 IJ
1 Dollar (Now York) .
96 Skillings (Copenhagen)
1 Lira (Venice)
1 Lira (Genoa)
1 Lira (Leghorn) .
1 Ruble . .
*. d.
4 3
2 2
8}
3
TV OF £1.
5 per cent 6perciu(l
1437618
1464303
1489812
1514107
16 •37245
15 69281
1580267
18 00255
16 19290
163741S
1661685
1671128
1686789
1701704
1715908
1729436
1742320
1764591
1766277
1777407
1788006
1798101
1807715
1816872
1825592
1300316
1321083
1340616
1369072
1376183
1392908
1408404
1423023
'1436814
1449824
146209«»
1473678
1484602
1494907
16046,10
1613801
1622454
1530617
1638318
1645583
1662437
1568903
1666002
1570767
1676186
Statute.
A.
B
32
1
48
2
64
3
80
3
161
3
323
3
485
3
647
3
809
3
619
3
P. T
23 14}
16 6.1
6 28j
38 20
37 10
34 21
32 2
29 123
26 23
13 16J
».
d.
,
. 4
3 1
jcn)
o
2
.
. U
8
.
,
9
•
.
7
«
. 3
li